Solved: In Exercises 4354, find an equation of the line

The simplest mathematical model for relating two variables is the ________ equation in two variables y mx b.I

For a line, the ratio of the change in to the change in is called the ________ of the line.

The ________-________ form of the equation of a line with slope passing through is y y1 mx x1.x1

Two lines are ________ if and only if their slopes are equal.

Two lines are ________ if and only if their slopes are negative reciprocals of each other.

When the -axis and -axis have different units of measure, the slope can be interpreted as a ________.

The prediction method ________ ________ is the method used to estimate a point on a line when the point does not lie between the given points.

Every line has an equation that can be written in ________ form.

In Exercises 9 and 10, identify the line that has each slope. m m 0 2 3 b) is undefined. m 2 m 1m

In Exercises 9 and 10, identify the line that has each slope. m m 0 2 3 m 3 m 2 m 1m

In Exercises 11 and 12, sketch the lines through the point with the indicated slopes on the same set of coordinate axes. 2, 3x

In Exercises 11 and 12, sketch the lines through the point with the indicated slopes on the same set of coordinate axes. 4, 1) 3

In Exercises 13 and 14, estimate the slope of the line.

In Exercises 13 and 14, estimate the slope of the line.

In Exercises 1524, find the slope and -intercept (if possible) of the equation of the line. Sketch the line. y 5x 3 y

In Exercises 1524, find the slope and -intercept (if possible) of the equation of the line. Sketch the line. y x 10

In Exercises 1524, find the slope and -intercept (if possible) of the equation of the line. Sketch the line. y x 6 12x 4y

In Exercises 1524, find the slope and -intercept (if possible) of the equation of the line. Sketch the line. y 32 y x 6 1

In Exercises 1524, find the slope and -intercept (if possible) of the equation of the line. Sketch the line. y 3 0

In Exercises 1524, find the slope and -intercept (if possible) of the equation of the line. Sketch the line. x 5 0

In Exercises 1524, find the slope and -intercept (if possible) of the equation of the line. Sketch the line. 5x 2 0

In Exercises 1524, find the slope and -intercept (if possible) of the equation of the line. Sketch the line. 3y 5 0

In Exercises 1524, find the slope and -intercept (if possible) of the equation of the line. Sketch the line. 7x 6y 30

In Exercises 1524, find the slope and -intercept (if possible) of the equation of the line. Sketch the line. 2x 3y 9

In Exercises 2534, plot the points and find the slope of the line passing through the pair of points. 0, 9, 6, 012,

In Exercises 2534, plot the points and find the slope of the line passing through the pair of points. 12, 0, 0, 87

In Exercises 2534, plot the points and find the slope of the line passing through the pair of points. 3, 2, 1, 62,

In Exercises 2534, plot the points and find the slope of the line passing through the pair of points. 2, 4, 4, 40

In Exercises 2534, plot the points and find the slope of the line passing through the pair of points. 5, 7, 8, 72,

In Exercises 2534, plot the points and find the slope of the line passing through the pair of points. 2, 1, 4, 5

In Exercises 2534, plot the points and find the slope of the line passing through the pair of points. 6, 1, 6, 40,

In Exercises 2534, plot the points and find the slope of the line passing through the pair of points. 0, 10, 4, 05,

In Exercises 2534, plot the points and find the slope of the line passing through the pair of points. 4.8, 3.1, 5.2, 1.66

In Exercises 2534, plot the points and find the slope of the line passing through the pair of points. 11 2 , 4 3, 3 2, 1

In Exercises 3542, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.) 2, 1, m 0 3,

In Exercises 3542, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.) 3, 2, m 0m

In Exercises 3542, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.)8, 1, m is undefined.

In Exercises 3542, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.) 1, 5, m is undefined.

In Exercises 3542, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.) 5, 4, m 21,

In Exercises 3542, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.) 0, 9, m 25

In Exercises 3542, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.) 2 1, 6,0

In Exercises 3542, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.) 2 7, 2m

In Exercises 4354, find an equation of the line that passes through the given point and has the indicated slope Sketch the line. 0, 2, m 3 0

In Exercises 4354, find an equation of the line that passes through the given point and has the indicated slope Sketch the line. 0, 10, m 1

In Exercises 4354, find an equation of the line that passes through the given point and has the indicated slope Sketch the line. 3, 6, m 2 0

In Exercises 4354, find an equation of the line that passes through the given point and has the indicated slope Sketch the line. 0, 0, m 40

In Exercises 4354, find an equation of the line that passes through the given point and has the indicated slope Sketch the line. 8, 2, 1 3 4, 0,

In Exercises 4354, find an equation of the line that passes through the given point and has the indicated slope Sketch the line. 4 m 8,

In Exercises 4354, find an equation of the line that passes through the given point and has the indicated slope Sketch the line. m 2, 5, 1 2 2, 3

In Exercises 4354, find an equation of the line that passes through the given point and has the indicated slope Sketch the line. 4 m 2, 5, 1 2

In Exercises 4354, find an equation of the line that passes through the given point and has the indicated slope Sketch the line. 6, 1, m m

In Exercises 4354, find an equation of the line that passes through the given point and has the indicated slope Sketch the line. 10, 4, m 6

In Exercises 4354, find an equation of the line that passes through the given point and has the indicated slope Sketch the line. 4, m 0

In Exercises 4354, find an equation of the line that passes through the given point and has the indicated slope Sketch the line. 5.1, 1.8, m 5 5

In Exercises 5564, find an equation of the line passing through the points. Sketch the line. 5, 1, 5, 54,

In Exercises 5564, find an equation of the line passing through the points. Sketch the line. 4, 3, 4, 4 4

In Exercises 5564, find an equation of the line passing through the points. Sketch the line.8, 1, 8, 71

In Exercises 5564, find an equation of the line passing through the points. Sketch the line. 1, 4, 6, 4 5

In Exercises 5564, find an equation of the line passing through the points. Sketch the line.2, 1 2, 1 2, 5 4

In Exercises 5564, find an equation of the line passing through the points. Sketch the line. 1, 1, 6, 2 3 2, 1 2

In Exercises 5564, find an equation of the line passing through the points. Sketch the line.1, 0.6, 2, 0.68

In Exercises 5564, find an equation of the line passing through the points. Sketch the line. 8, 0.6, 2, 2.4 1,

In Exercises 5564, find an equation of the line passing through the points. Sketch the line. 2, 1, , 11 3, 1 1

In Exercises 5564, find an equation of the line passing through the points. Sketch the line. 7 3, 8, 7 3 2, 1, , 11 3,

In Exercises 6568, determine whether the lines are parallel, perpendicular, or neither. L1: y : y 4x 1 1 3 x 2 2: y : y 4x 7 1 3 x 3 L1

In Exercises 6568, determine whether the lines are parallel, perpendicular, or neither. L1 L1: y : y 4x 1 1 L2 L2: y : y 4x 7 1

In Exercises 6568, determine whether the lines are parallel, perpendicular, or neither. L1: y x 5 1 2 x 3 L2: y x 1 1 2 x 1 L

In Exercises 6568, determine whether the lines are parallel, perpendicular, or neither. L1: y 4 5 L1: y x 5 1 L2: y 5 4 L2: y x 1 1

In Exercises 6972, determine whether the lines and passing through the pairs of points are parallel, perpendicular, or neither. L : 2, 1, 1, 51: 0, 1, 5, 9 L1 L2 L : 1, 3, 5, 52: 0, 3, 4, 1 L1 L

In Exercises 6972, determine whether the lines and passing through the pairs of points are parallel, perpendicular, or neither. L1 L : 2, 1, 1, 51: 0 L2 L : 1, 3, 5, 52: 0

In Exercises 6972, determine whether the lines and passing through the pairs of points are parallel, perpendicular, or neither. L : 4, 8, 4, 21: 3, 6, 6, 0 L2 L : 2: 0, 1, 5, 7 3

In Exercises 6972, determine whether the lines and passing through the pairs of points are parallel, perpendicular, or neither. L1 L : 4, 8, 4, 21: 3 L2: 3, 5, 1, 1 3 L 2: 0

In Exercises 7380, write equations of the lines through the given point (a) parallel to and (b) perpendicular to the given line. 4x 2y 3, 2, 1 L2

In Exercises 7380, write equations of the lines through the given point (a) parallel to and (b) perpendicular to the given line. x y 7, 3, 2 4x

In Exercises 7380, write equations of the lines through the given point (a) parallel to and (b) perpendicular to the given line. 8 3x 4y 7, x

In Exercises 7380, write equations of the lines through the given point (a) parallel to and (b) perpendicular to the given line. 4 5x 3y 0, 2

In Exercises 7380, write equations of the lines through the given point (a) parallel to and (b) perpendicular to the given line. y 3 0, 1, 0 7

In Exercises 7380, write equations of the lines through the given point (a) parallel to and (b) perpendicular to the given line. x 4 0, 3, 2 y

In Exercises 7380, write equations of the lines through the given point (a) parallel to and (b) perpendicular to the given line. x y 4, 2.5, 6.8 x

In Exercises 7380, write equations of the lines through the given point (a) parallel to and (b) perpendicular to the given line.6x 2y 9, 3.9, 1.4 x 

In Exercises 8186, use the intercept form to find the equation of the line with the given intercepts. The intercept form of the equation of a line with intercepts and is 81. -intercept: x 2, 0 a y 0, 3 x

In Exercises 8186, use the intercept form to find the equation of the line with the given intercepts. The intercept form of the equation of a line with intercepts and is x 3, 0 y y 0, 4 x

In Exercises 8186, use the intercept form to find the equation of the line with the given intercepts. The intercept form of the equation of a line with intercepts and is 6 x , 0 0, 2

In Exercises 8186, use the intercept form to find the equation of the line with the given intercepts. The intercept form of the equation of a line with intercepts and is 3 x , 0 y 0, 2 2

In Exercises 8186, use the intercept form to find the equation of the line with the given intercepts. The intercept form of the equation of a line with intercepts and is Point on line: 1, 2 x c, 0 1 y 0, c, c 0 x

In Exercises 8186, use the intercept form to find the equation of the line with the given intercepts. The intercept form of the equation of a line with intercepts and is86. Point on line: 3, 4 y x d, 0  y 0, d, d 0 x

Sales The following are the slopes of lines representing annual sales in terms of time in years. Use the slopes to interpret any change in annual sales for a one-year increase in time. (a) The line has a slope of (b) The line has a slope of (c) The line has a slope of

Sales The graph shows the sales (in billions of dollars) for Apple Inc. in the years 2004 through 2010. (Source: Apple Inc.) (a) Use the slopes of the line segments to determine the years in which the sales showed the greatest increase and the least increase. (b) Find the slope of the line segment connecting the points for the years 2004 and 2010. (c) Interpret the meaning of the slope in part (b) in the context of the problem.

Road Grade You are driving on a road that has a 6% uphill grade. This means that the slope of the road is Approximate the amount of vertical change in your position when you drive 200 feet.

From the top of a mountain road, a surveyor takes several horizontal measurements and several vertical measurements as shown in the table ( and are measured in feet). (a) Sketch a scatter plot of the data. (b) Use a straightedge to sketch the line that you think best fits the data. (c) Find an equation for the line you sketched in part (b). (d) Interpret the meaning of the slope of the line in part (c) in the context of the problem. (e) The surveyor needs to put up a road sign that indicates the steepness of the road. For instance, a surveyor would put up a sign that states 8% grade on a road with a downhill grade that has a slope of What should the sign state for the road in this problem?

In Exercises 91 and 92, you are given the dollar value of a product in 2013 and the rate at which the value of the product is expected to change during the next 5 years. Use this information to write a linear equation that gives the dollar value of the product in terms of the year (Let represent 2013.) 91. $2540 $125 decrease per year

In Exercises 91 and 92, you are given the dollar value of a product in 2013 and the rate at which the value of the product is expected to change during the next 5 years. Use this information to write a linear equation that gives the dollar value of the product in terms of the year (Let represent 2013.) 92. $156 $4.50 increase per year

Cost The cost of producing computer laptop bags is given by Explain what the -intercept and the slope measure.

Monthly Salary A pharmaceutical salesperson receives a monthly salary of $2500 plus a commission of 7% of sales. Write a linear equation for the salespersons monthly wage in terms of monthly sales S

Depreciation A sub shop purchases a used pizza oven for $875. After 5 years, the oven will have to be discarded and replaced. Write a linear equation giving the value of the equipment during the 5 years it will be in use

Depreciation A school district purchases a high-volume printer, copier, and scanner for $24,000. After 10 years, the equipment will have to be replaced. Its value at that time is expected to be $2000. Write a linear equation giving the value of the equipment during the 10 years it will be in use

Temperature Conversion Write a linear equation that expresses the relationship between the temperature in degrees Celsius and degrees Fahrenheit Use the fact that water freezes at and boils at 100C 212F.0

Brain Weight The average weight of a male childs brain is 970 grams at age 1 and 1270 grams at age 3. (Source: American Neurological Association) (a) Assuming that the relationship between brain weight and age is linear, write a linear model for the data. (b) What is the slope and what does it tell you about brain weight? (c) Use your model to estimate the average brain weight at age 2. (d) Use your schools library, the Internet, or some other reference source to find the actual average brain weight at age 2. How close was your estimate? (e) Do you think your model could be used to determine the average brain weight of an adult? Explain

Cost, Revenue, and Profit A roofing contractor purchases a shingle delivery truck with a shingle elevator for $42,000. The vehicle requires an average expenditure of $9.50 per hour for fuel and maintenance, and the operator is paid $11.50 per hour. (a) Write a linear equation giving the total cost of operating this equipment for hours. (Include the purchase cost of the equipment.) (b) Assuming that customers are charged $45 per hour of machine use, write an equation for the revenue derived from hours of use. (c) Use the formula for profit to write an equation for the profit derived from hours of use. (d) Use the result of part (c) to find the break-even pointthat is, the number of hours this equipment must be used to yield a profit of 0 dollars.

Geometry The length and width of a rectangular garden are 15 meters and 10 meters, respectively. A walkway of width surrounds the garden. (a) Draw a diagram that gives a visual representation of the problem. (b) Write the equation for the perimeter of the walkway in terms of (c) Use a graphing utility to graph the equation for the perimeter. (d) Determine the slope of the graph in part (c). For each additional one-meter increase in the width of the walkway, determine the increase in its perimeter.

In Exercises 101 and 102, determine whether the statement is true or false. Justify your answer. A line with a slope of is steeper than a line with a slope of

In Exercises 101 and 102, determine whether the statement is true or false. Justify your answer. The line through and and the line through and are parallel.

Right Triangle Explain how you could use slope to show that the points and are the vertices of a right triangle

Vertical Line Explain why the slope of a vertical line is said to be undefined.

Think About It With the information shown in the graphs, is it possible to determine the slope of each line? Is it possible that the lines could have the same slope? Explain.

Perpendicular Segments Find and in terms of and respectively (see figure). Then use the Pythagorean Theorem to find a relationship between and

Think About It Is it possible for two lines with positive slopes to be perpendicular? Explain.

Slope and Steepness The slopes of two lines are and Which is steeper? Explain.

Comparing Slopes Use a graphing utility to compare the slopes of the lines where 1, 2, and 4. Which line rises most quickly? Now, let and Which line falls most quickly? Use a square setting to obtain a true geometric perspective. What can you conclude about the slope and the rate at which the line rises or falls?

HOW DO YOU SEE IT? Match the description of the situation with its graph. Also determine the slope and -intercept of each graph and interpret the slope and -intercept in the context of the situation. [The graphs are labeled (i), (ii), (iii), and (iv).] (i) (ii) (iii) (iv) (a) A person is paying $20 per week to a friend to repay a $200 loan. (b) An employee receives $12.50 per hour plus $2 for each unit produced per hour. (c) A sales representative receives $30 per day for food plus $0.32 for each mile traveled. (d) A computer that was purchased for $750 depreciates $100 per year.

111114, find a relationship between and such that is equidistant (the same distance) from the two points 4, 1, 2, 36, 5, 1, 8 x, y

111114, find a relationship between and such that is equidistant (the same distance) from the two points 6, 5, 1, 8 

111114, find a relationship between and such that is equidistant (the same distance) from the two points 3, 5 2, 7, 1 4

111114, find a relationship between and such that is equidistant (the same distance) from the two points 2, 4, 7 2, 5 4 3, 5 2

A relation that assigns to each element from a set of inputs, or ________, exactly one element in a set of outputs, or ________, is called a ________.

For an equation that represents as a function of the set of all values taken on by the ________ variable is the domain, and the set of all values taken on by the ________ variable is the range.

If the domain of the function is not given, then the set of values of the independent variable for which the expression is defined is called the ________ ________.

. In calculus, one of the basic definitions is that of a ________ ________, given by fx hfx h

In Exercises 58, determine whether the relation represents as a function of x

In Exercises 58, determine whether the relation represents as a function of

In Exercises 58, determine whether the relation represents as a function of

In Exercises 58, determine whether the relation represents as a function of

0, 2, 3, 0, 1, 1 0, 0, 1, 0, 2, 0, 3, 0 0, 1, 2, 2, 1, 2, 3, 0, 1, 1 0, 1, 1, 2, 2, 0, 3, 2 A 0, 1, 2, 3B 2, 1, 0,

1, a, 0, a, 2, c, 3, b a, 1, b, 2, c, 3 a, 1, c, 2, c, 3, b, 3 A a, b, cB 0,

In Exercises 1120, determine whether the equation represents as a function of x x 2 y 4 2 y 2 4

In Exercises 1120, determine whether the equation represents as a function of x x x 2 y 4

In Exercises 1120, determine whether the equation represents as a function of x 2x 3y 4

In Exercises 1120, determine whether the equation represents as a function of x x 22 y 2x 3y 4 2 4x

In Exercises 1120, determine whether the equation represents as a function of x y 16 x y x 5 2 x

In Exercises 1120, determine whether the equation represents as a function of x y x 5

In Exercises 1120, determine whether the equation represents as a function of x y 4 x 4 x y

In Exercises 1120, determine whether the equation represents as a function of xy y 4 x 4

In Exercises 1120, determine whether the equation represents as a function of x y 75 x

In Exercises 1120, determine whether the equation represents as a function of x x 1 0

In Exercises 2132, evaluate (if possible) the function at each specified value of the independent variable and simplify fx2x 3 y f1f3fx 1 fx2

In Exercises 2132, evaluate (if possible) the function at each specified value of the independent variable and simplifyVr4 3r3 (a) (b) (c)

In Exercises 2132, evaluate (if possible) the function at each specified value of the independent variable and simplifygt4t 2 3t 5 V g2gt 2gtg2 gt4t

In Exercises 2132, evaluate (if possible) the function at each specified value of the independent variable and simplify htt2 2t g (a) (b) (c)

In Exercises 2132, evaluate (if possible) the function at each specified value of the independent variable and simplify fy3 y h f4x2 f4f0.25 fy

In Exercises 2132, evaluate (if possible) the function at each specified value of the independent variable and simplify xx 8 2 f8f1fx 8 fx

In Exercises 2132, evaluate (if possible) the function at each specified value of the independent variable and simplify qx1 x2 9 f8 q0q3qy 3 qx

In Exercises 2132, evaluate (if possible) the function at each specified value of the independent variable and simplify qt2t 2 2 3 3 t 2 q2q0qx qt

In Exercises 2132, evaluate (if possible) the function at each specified value of the independent variable and simplify f2f2fx 1 fxx

In Exercises 2132, evaluate (if possible) the function at each specified value of the independent variable and simplify fxx4 f (a) (b) (c)

In Exercises 2132, evaluate (if possible) the function at each specified value of the independent variable and simplify fx 2 2x 1, x < 0 2x 2, x 0 f1f0f2 fx

In Exercises 2132, evaluate (if possible) the function at each specified value of the independent variable and simplify fx 4 0, 4 5x, x2 1, x2 1, x 2 2 < x < 2 x 2

Evaluating a Function In Exercises 3336, complete the table. fxx2 3 97

Evaluating a Function In Exercises 3336, complete the table. ht1 2t 3

Evaluating a Function In Exercises 3336, complete the table. fx 1 2x 4, x 0 x 22, x > 0 ht

Evaluating a Function In Exercises 3336, complete the table. x 9 x 2, x < 3 x 3, x 3 fx

In Exercises 3744, find all real values of such that x fx0. fx15 3x fx

In Exercises 3744, find all real values of such that x fx0. fx3x 4 5

In Exercises 3744, find all real values of such that x fx0. xx 2 9 f

In Exercises 3744, find all real values of such that x fx0.fxx2 8x 15 f

In Exercises 3744, find all real values of such that x fx0. fxx3 x f

In Exercises 3744, find all real values of such that x fx0. fxx3 x2 4x 4 fx

In Exercises 3744, find all real values of such that x fx0. fx5x 1

In Exercises 3744, find all real values of such that x fx0. fx12 x2

In Exercises 4548, find the value(s) of for which x fxgx. fxx gxx 2 2, x f

In Exercises 4548, find the value(s) of for which x fxgx. fxx gx7x 5 2 2x 1, fxx

In Exercises 4548, find the value(s) of for which x fxgx. gx2x2 fxx4 2x 2 , f

In Exercises 4548, find the value(s) of for which x fxgx. fxx 4, gx2 x gx2x

In Exercises 4960, find the domain of the function. fx5x2 2x 1 f

In Exercises 4960, find the domain of the function. x1 2x2 fx

In Exercises 4960, find the domain of the function. ht4

In Exercises 4960, find the domain of the function. sy3y

In Exercises 4960, find the domain of the function. gyy 10 t

In Exercises 4960, find the domain of the function. t 3 gyy 10 t 4 sy

In Exercises 4960, find the domain of the function.x1 x 3 x 2 f

In Exercises 4960, find the domain of the function.hx10 x 2 2x g

In Exercises 4960, find the domain of the function. fss 1 s 4 hx

In Exercises 4960, find the domain of the function. fxx 6 6 x f

In Exercises 4960, find the domain of the function.xx 4 

In Exercises 4960, find the domain of the function. fxx 2 x 10 fx

Maximum Volume An open box of maximum volume is to be made from a square piece of material 24 centimeters on a side by cutting equal squares from the corners and turning up the sides (see figure). (a) The table shows the volumes (in cubic centimeters) of the box for various heights (in centimeters). Use the table to estimate themaximum volume. (b) Plot the points from the table in part (a). Does the relation defined by the ordered pairs represent as a function of (c) Given that is a function of write the function and determine its domain.

Maximum Profit The cost per unit in the production of an MP3 player is $60. The manufacturer charges $90 per unit for orders of 100 or less. To encourage large orders, the manufacturer reduces the charge by $0.15 per MP3 player for each unit ordered in excess of 100 (for example, there would be a charge of $87 per MP3 player for an order size of 120). (a) The table shows the profits (in dollars) for various numbers of units ordered, Use the table to estimate the maximum profit. (b) Plot the points from the table in part (a). Does the relation defined by the ordered pairs represent as a function of (c) Given that is a function of write the function and determine its domain. (Note: where is revenue and is cost.)

Geometry Write the area of a square as a function of its perimeter

Geometry Write the area of a circle as a function of its circumference

Path of a Ball The height (in feet) of a baseball thrown by a child is where is the horizontal distance (in feet) from where the ball was thrown. Will the ball fly over the head of another child 30 feet away trying to catch the ball? (Assume that the child who is trying to catch the ball holds a baseball glove at a height of 5 feet.)

Postal Regulations A rectangular package to be sent by the U.S. Postal Service can have a maximum combined length and girth (perimeter of a cross section) of 108 inches (see figure). (a) Write the volume of the package as a function of What is the domain of the function? (b) Use a graphing utility to graph the function. Be sure to use an appropriate window setting. (c) What dimensions will maximize the volume of the package? Explain your answer

Geometry A right triangle is formed in the first quadrant by the - and -axes and a line through the point (see figure). Write the area of the triangle as a function of and determine the domain of the function. Figure for 67 Figure for 68 68. Geometry A rectangle is bounded by the -axis and the semicircle (see figure). Write the area of the rectangle as a function of and graphically determine the domain of the function.

Prescription Drugs The percents of prescriptions filled with generic drugs in the United States from 2004 through 2010 (see figure) can be approximated by the model where represents the year, with corresponding to 2004. Use this model to find the percent of prescriptions filled with generic drugs in each year from 2004 through 2010. (Source: National Association of Chain Drug Stores)

Median Sale Price The median sale prices (in thousands of dollars) of an existing one-family home in the United States from 2000 through 2010 (see figure) can be approximated by the model where represents the year, with corresponding to 2000. Use this model to find the median sale price of an existing one-family home in each year from 2000 through 2010. (Source: National Association of Realtors)

Cost, Revenue, and Profit A company produces a product for which the variable cost is $12.30 per unit and the fixed costs are $98,000. The product sells for $17.98. Let be the number of units produced and sold. (a) The total cost for a business is the sum of the variable cost and the fixed costs. Write the total cost as a function of the number of units produced. (b) Write the revenue as a function of the number of units sold. (c) Write the profit as a function of the number of units sold. (Note: )

Cost, Revenue, and Profit A company produces a product for which the variable cost is $12.30 per unit and the fixed costs are $98,000. The product sells for $17.98. Let be the number of units produced and sold. (a) The total cost for a business is the sum of the variable cost and the fixed costs. Write the total cost as a function of the number of units produced. (b) Write the revenue as a function of the number of units sold. (c) Write the profit as a function of the number of units sold. (Note: )

Average Cost The inventor of a new game believes that the variable cost for producing the game is $0.95 per unit and the fixed costs are $6000. The inventor sells each game for $1.69. Let be the number of games sold. (a) The total cost for a business is the sum of the variable cost and the fixed costs. Write the total cost as a function of the number of games sold. (b) Write the average cost per unit as a function of x

Height of a Balloon A balloon carrying a transmitter ascends vertically from a point 3000 feet from the receiving station. (a) Draw a diagram that gives a visual representation of the problem. Let represent the height of the balloon and let represent the distance between the balloon and the receiving station. (b) Write the height of the balloon as a function of What is the domain of the function?

The function estimates the force (in tons) of water against the face of a dam, where is the depth of the water (in feet). (a) Complete the table. What can you conclude from the table? (b) Use the table to approximate the depth at which the force against the dam is 1,000,000 tons. (c) Find the depth at which the force against the dam is 1,000,000 tons algebraically.

Transportation For groups of 80 or more people, a charter bus company determines the rate per person according to the formula where the rate is given in dollars and is the number of people. (a) Write the revenue for the bus company as a function of (b) Use the function in part (a) to complete the table. What can you conclude?

E-Filing The table shows the numbers of tax returns (in millions) made through e-file from 2003 through 2010. Let represent the number of tax returns made through e-file in the year (Source: Internal Revenue Service) (a) Find and interpret the result in the context of the problem. (b) Make a scatter plot of the data. (c) Find a linear model for the data algebraically. Let represent the number of tax returns made through e-file and let correspond to 2003. (d) Use the model found in part (c) to complete the table. (e) Compare your results from part (d) with the actual data. (f) Use a graphing utility to find a linear model for the data. Let correspond to 2003. How does the model you found in part (c) compare with the model given by the graphing utility?

In Exercises 7784, find the difference quotient and simplify your answer. h fxx , h 0 2 x 1, 97

In Exercises 7784, find the difference quotient and simplify your answer. h fx5x x , h 0 2, f2

In Exercises 7784, find the difference quotient and simplify your answer. h fxx , h 0 3 3x, f

In Exercises 7784, find the difference quotient and simplify your answer.h fx4x , h 0 2 2x, fx

In Exercises 7784, find the difference quotient and simplify your answer. h fx4x , h 0 2 2x, fx

In Exercises 7784, find the difference quotient and simplify your answer. t 1 ft, t 1 1

In Exercises 7784, find the difference quotient and simplify your answer. x 5 fx5x, , ft

In Exercises 7784, find the difference quotient and simplify your answer. x 8 fxx , 2 3 x 8 fxf8 x

In Exercises 8588, match the data with one of the following functions and determine the value of the constant that will make the function fit the data in the table.

In Exercises 8588, match the data with one of the following functions and determine the value of the constant that will make the function fit the data in the table.

In Exercises 8588, match the data with one of the following functions and determine the value of the constant that will make the function fit the data in the table.

In Exercises 8588, match the data with one of the following functions and determine the value of the constant that will make the function fit the data in the table.

In Exercises 8992, determine whether the statement is true or false. Justify your answer. 89. Every relation is a function.

In Exercises 8992, determine whether the statement is true or false. Justify your answer. 90. Every function is a relation.

In Exercises 8992, determine whether the statement is true or false. Justify your answer. For the function the domain is

In Exercises 8992, determine whether the statement is true or false. Justify your answer. The set of ordered pairs represents a function. f fxx2 gf gx 3 fxx 2 x 2. gf gx1 x 1 fxx 1 . 2, 2, 0, 4, 2, 2 8, 2, 6, 0, 4, 0, , 0, . fxx

Think About It Consider x 1 fxx 1 . 2, Why are the domains of and different?

Think About It Consider 3 fxx 2 x 2. gf

Think About It Given is the independent variable? Why or why not?

HOW DO YOU SEE IT? The graph represents the height of a projectile after seconds. (a) Explain why is a function of (b) Approximate the height of the projectile after 0.5 second and after 1.25 seconds. (c) Approximate the domain of (d) Is a function of Explain. t h? h. h t. t 5 10 15 20 25 30 h 0.5 1.0 1.5 2.0 2.5 Time, t (in seconds) H

Think About It In Exercises 97 and 98, determine whether the statements use the word function in ways that are mathematically correct. Explain your reasoning (a) The sales tax on a purchased item is a function of the selling price. b) Your score on the next algebra exam is a function of the number of hours you study the night before the exam

Think About It In Exercises 97 and 98, determine whether the statements use the word function in ways that are mathematically correct. Explain your reasoning(a) The amount in your savings account is a function of your salary. (b) The speed at which a free-falling baseball strikes the ground is a function of the height from which itwas dropped.

The ________ ________ ________ is used to determine whether the graph of an equation is a function of in terms of x

The ________ of a function are the values of for which f x fx0.x.

A function is ________ on an interval when, for any and in the interval fx1> fx2 x . 1 < x2 x2 x1 ff x

A function value is a relative ________ of when there exists an interval containing such that x fafx. 1 < x < x2faf

The ________ ________ ________ ________ between any two points and is the slope of the line through the two points, and this line is called the ________ line.

A function is ________ when, for each in the domain of f x f, fx fx. x2, fx

In Exercises 710, use the graph of the function to find the domain and range of and the indicated function values. f2f1f f1f1f ff1f0f112f2f1f1

In Exercises 710, use the graph of the function to find the domain and range of and the indicated function values. f1f2 f2f f0f1 f11

In Exercises 710, use the graph of the function to find the domain and range of and the indicated function values. f2f1f1f2 f3f f1f0

In Exercises 710, use the graph of the function to find the domain and range of and the indicated function values. f2f1 f1 f0f f2

In Exercises 1114, use the Vertical Line Test to determine whether is a function of To print an enlarged copy of the graph, go to MathGraphs.com. y 2 1 1 4x3

In Exercises 1114, use the Vertical Line Test to determine whether is a function of To print an enlarged copy of the graph, go to MathGraphs.com. x y y 2 1 1

In Exercises 1114, use the Vertical Line Test to determine whether is a function of To print an enlarged copy of the graph, go to MathGraphs.com. x 2 2xy 1 2 y 2 25 x

In Exercises 1114, use the Vertical Line Test to determine whether is a function of To print an enlarged copy of the graph, go to MathGraphs.com. 2 2xy 1 2

In Exercises 1524, find the zeros of the function algebraically. fx2x2 7x 30 x 2

In Exercises 1524, find the zeros of the function algebraically. fx3x2 22x 16 fx2

In Exercises 1524, find the zeros of the function algebraically. fxx 9x

In Exercises 1524, find the zeros of the function algebraically. fxx2 9x 14 4x

In Exercises 1524, find the zeros of the function algebraically. fx1 2 x3 x fx

In Exercises 1524, find the zeros of the function algebraically.fx9x4 25x2 fx

In Exercises 1524, find the zeros of the function algebraically. fxx3 4x2 9x 36 fx9

In Exercises 1524, find the zeros of the function algebraically. fx4x3 24x2 x 6 fxx

In Exercises 1524, find the zeros of the function algebraically. fx2x 1 fx

In Exercises 1524, find the zeros of the function algebraically. fx3x 2 f

In Exercises 2530, (a) use a graphing utility to graph the function and find the zeros of the function and (b) verify your results from part (a) algebraically. fx3 fxxx 75 x fx3x 

In Exercises 2530, (a) use a graphing utility to graph the function and find the zeros of the function and (b) verify your results from part (a) algebraically. fxxx 75 x

In Exercises 2530, (a) use a graphing utility to graph the function and find the zeros of the function and (b) verify your results from part (a) algebraically. fx2x 11 fx

In Exercises 2530, (a) use a graphing utility to graph the function and find the zeros of the function and (b) verify your results from part (a) algebraically. fx3x 14 8 fx3

In Exercises 2530, (a) use a graphing utility to graph the function and find the zeros of the function and (b) verify your results from part (a) algebraically. fx3x 1 x 6 fx

In Exercises 2530, (a) use a graphing utility to graph the function and find the zeros of the function and (b) verify your results from part (a) algebraically. fx2x2 9 3 x fx3x

In Exercises 3138, determine the intervals on which the function is increasing, decreasing, or constant. fx2 4x 3 2x 97

In Exercises 3138, determine the intervals on which the function is increasing, decreasing, or constant. fxx fx2 4x 3 2x

In Exercises 3138, determine the intervals on which the function is increasing, decreasing, or constant.fxx 2 1 3 3x2 2 x

In Exercises 3138, determine the intervals on which the function is increasing, decreasing, or constant. fxx fxx 2 1 3 3x2

In Exercises 3138, determine the intervals on which the function is increasing, decreasing, or constant. fxx 1x 1 x

In Exercises 3138, determine the intervals on which the function is increasing, decreasing, or constant. fxx2 x 1 x 1 fx

In Exercises 3138, determine the intervals on which the function is increasing, decreasing, or constant. fx x 3, 3, 2x 1, x 0 0 < x 2 x > 2 x

In Exercises 3138, determine the intervals on which the function is increasing, decreasing, or constant. fx 2x 1, x2 2, x 1 x > 1 x 2

In Exercises 3946, (a) use a graphing utility to graph the function and visually determine the intervals on which the function is increasing, decreasing, or constant, and (b) make a table of values to verify whether the function is increasing, decreasing, or constant on the intervals you identified in part (a). fx3 gx

In Exercises 3946, (a) use a graphing utility to graph the function and visually determine the intervals on which the function is increasing, decreasing, or constant, and (b) make a table of values to verify whether the function is increasing, decreasing, or constant on the intervals you identified in part (a). gxx x

In Exercises 3946, (a) use a graphing utility to graph the function and visually determine the intervals on which the function is increasing, decreasing, or constant, and (b) make a table of values to verify whether the function is increasing, decreasing, or constant on the intervals you identified in part (a). gss2 4 f

In Exercises 3946, (a) use a graphing utility to graph the function and visually determine the intervals on which the function is increasing, decreasing, or constant, and (b) make a table of values to verify whether the function is increasing, decreasing, or constant on the intervals you identified in part (a). fx3x4 6x2 gss

In Exercises 3946, (a) use a graphing utility to graph the function and visually determine the intervals on which the function is increasing, decreasing, or constant, and (b) make a table of values to verify whether the function is increasing, decreasing, or constant on the intervals you identified in part (a). fx1 x fxx

In Exercises 3946, (a) use a graphing utility to graph the function and visually determine the intervals on which the function is increasing, decreasing, or constant, and (b) make a table of values to verify whether the function is increasing, decreasing, or constant on the intervals you identified in part (a). fxxx 3 f

In Exercises 3946, (a) use a graphing utility to graph the function and visually determine the intervals on which the function is increasing, decreasing, or constant, and (b) make a table of values to verify whether the function is increasing, decreasing, or constant on the intervals you identified in part (a).fxx 32 f

In Exercises 3946, (a) use a graphing utility to graph the function and visually determine the intervals on which the function is increasing, decreasing, or constant, and (b) make a table of values to verify whether the function is increasing, decreasing, or constant on the intervals you identified in part (a). fxx23 fx

In Exercises 4754, use a graphing utility to graph the function and approximate (to two decimal places) any relative minima or maxima. fx3x 2 3x 2 2 2x 5 fxx2

In Exercises 4754, use a graphing utility to graph the function and approximate (to two decimal places) any relative minima or maxima. fx x fx3x 2 3x 2 2 2x 5

In Exercises 4754, use a graphing utility to graph the function and approximate (to two decimal places) any relative minima or maxima. fx 2x fxxx 2x 32 9x fx x fx3x

In Exercises 4754, use a graphing utility to graph the function and approximate (to two decimal places) any relative minima or maxima. fxxx 2x 32 9x f

In Exercises 4754, use a graphing utility to graph the function and approximate (to two decimal places) any relative minima or maxima. fxx3 3x2 x 1 fx 2

In Exercises 4754, use a graphing utility to graph the function and approximate (to two decimal places) any relative minima or maxima. hxx3 6x2 15 fx

In Exercises 4754, use a graphing utility to graph the function and approximate (to two decimal places) any relative minima or maxima. hxx 1x gxx4

In Exercises 4754, use a graphing utility to graph the function and approximate (to two decimal places) any relative minima or maxima. gxx4 x hx

In Exercises 5560, graph the function and determine the interval(s) for which f x 0. fx4 x fx4

In Exercises 5560, graph the function and determine the interval(s) for which f x 0. fx4x 2 f

In Exercises 5560, graph the function and determine the interval(s) for which f x 0. fx9 x 2 4x 2 fx4

In Exercises 5560, graph the function and determine the interval(s) for which f x 0. fxx fx9 x 2 4x 2 fx4 x4 x fx4

In Exercises 5560, graph the function and determine the interval(s) for which f x 0. fxx 1 f

In Exercises 5560, graph the function and determine the interval(s) for which f x 0. fx 1 xfxx

In Exercises 6164, find the average rate of change of the function from 2 x x1 0, x fx 2x 15 2 3 x2 x

In Exercises 6164, find the average rate of change of the function from 2 x x1 1, x fxx 2 5 2 2x 8 x1 0,

In Exercises 6164, find the average rate of change of the function from 2 x x1 1, x fxx 2 3 3 3x2 x x1 1,

In Exercises 6164, find the average rate of change of the function from 2 x x1 1, x fx x 2 6 3 6x2 x x1 1,

Research and Development The amounts (in millions of dollars) the U.S. Department of Energy spent for research and development from 2005 through 2010 can be approximated by the model where represents the year, with corresponding to 2005. (Source: American Association for the Advancement of Science) (a) Use a graphing utility to graph the model. (b) Find the average rate of change of the model from 2005 to 2010. Interpret your answer in the context of the problem.

Finding Average Speed Use the information in Example 7 to find the average speed of the car from to seconds. Explain why the result is less than the value obtained in part (b) of Example 7.

In Exercises 6770, (a) use the position equation to write a function that represents the situation, (b) use a graphing utility to graph the function, (c) find the average rate of change of the function from to (d) describe the slope of the secant line through and (e) find the equation of the secant line through and and (f) graph the secant line in the same viewing window as your position function. An object is thrown upward from a height of 6 feet at a velocity of 64 feet per second. t1 0, t2 3 t

In Exercises 6770, (a) use the position equation to write a function that represents the situation, (b) use a graphing utility to graph the function, (c) find the average rate of change of the function from to (d) describe the slope of the secant line through and (e) find the equation of the secant line through and and (f) graph the secant line in the same viewing window as your position function. An object is thrown upward from a height of 6.5 feet at a velocity of 72 feet per second. t1 0, t2 4 t

In Exercises 6770, (a) use the position equation to write a function that represents the situation, (b) use a graphing utility to graph the function, (c) find the average rate of change of the function from to (d) describe the slope of the secant line through and (e) find the equation of the secant line through and and (f) graph the secant line in the same viewing window as your position function. An object is thrown upward from ground level at a velocity of 120 feet per second t1 3, t2 5 t

In Exercises 6770, (a) use the position equation to write a function that represents the situation, (b) use a graphing utility to graph the function, (c) find the average rate of change of the function from to (d) describe the slope of the secant line through and (e) find the equation of the secant line through and and (f) graph the secant line in the same viewing window as your position function. An object is dropped from a height of 80 feet t1 1, t2 2 t

In Exercises 7176, determine whether the function is even, odd, or neither. Then describe the symmetry. fxx 3 5x 6 2x2 3 t1

In Exercises 7176, determine whether the function is even, odd, or neither. Then describe the symmetry. gxx fxx 3 5x 6 2x2

In Exercises 7176, determine whether the function is even, odd, or neither. Then describe the symmetry. fxx1 x hxxx 5 2 gxx f

In Exercises 7176, determine whether the function is even, odd, or neither. Then describe the symmetry. hxxx 5 2

In Exercises 7176, determine whether the function is even, odd, or neither. Then describe the symmetry. fs4s32 f

In Exercises 7176, determine whether the function is even, odd, or neither. Then describe the symmetry. gs4s 23 f

In Exercises 7782, sketch a graph of the function and determine whether it is even, odd, or neither. Verify your answer algebraically. fx 9 fx5

In Exercises 7782, sketch a graph of the function and determine whether it is even, odd, or neither. Verify your answer algebraically. fx5 3x gs

In Exercises 7782, sketch a graph of the function and determine whether it is even, odd, or neither. Verify your answer algebraically. fx x 52 4

In Exercises 7782, sketch a graph of the function and determine whether it is even, odd, or neither. Verify your answer algebraically. hxx fx x 52 4 fx 9 fx

In Exercises 7782, sketch a graph of the function and determine whether it is even, odd, or neither. Verify your answer algebraically. fx1 x t 1

In Exercises 7782, sketch a graph of the function and determine whether it is even, odd, or neither. Verify your answer algebraically. gt 3 fx1 x t 1 hxx fx

In Exercises 83 and 84, write the height of the rectangle as a function of x

In Exercises 83 and 84, write the height of the rectangle as a function of x

In Exercises 85 and 86, write the length of the rectangle as a function of y

In Exercises 85 and 86, write the length of the rectangle as a function of y

Lumens The number of lumens (time rate of flow of light) from a fluorescent lamp can be approximated by the model where is the wattage of the lamp. (a) Use a graphing utility to graph the function. (b) Use the graph from part (a) to estimate the wattage necessary to obtain 2000 lumens.

Geometry Corners of equal size are cut from a square with sides of length 8 meters (see figure). (a) Write the area of the resulting figure as a function of Determine the domain of the function. (b) Use a graphing utility to graph the area function over its domain. Use the graph to find the range of the function. (c) Identify the figure that results when is the maximum value in the domain of the function. What would be the length of each side of the figure?

Coordinate Axis Scale Each function described below models the specified data for the years 2003 through 2013, with corresponding to 2003. Estimate a reasonable scale for the vertical axis (e.g., hundreds, thousands, millions, etc.) of the graph and justify your answer. (There are many correct answers.) (a) represents the average salary of college professors. (b) represents the U.S. population. (c) represents the percent of the civilian work force that is unemployed.

The table shows the temperatures (in degrees Fahrenheit) in a city over a 24-hour period. Let represent the time of day, where corresponds to 6 A.M. A model that represents these data is given by (a) Use a graphing utility to create a scatter plot of the data. Then graph the model in the same viewing window. (b) How well does the model fit the data? (c) Use the graph to approximate the times when the temperature was increasing and decreasing. (d) Use the graph to approximate the maximum and minimum temperatures during this 24-hour period. (e) Could this model predict the temperatures in the city during the next 24-hour period? Why or why not?

Writing Use a graphing utility to graph each function. Write a paragraph describing any similarities and differences you observe among the graphs. y x y x 2 y x3 y x 4 y x5 y x 6

HOW DO YOU SEE IT? Use the graph of the function to answer (a)(e). (a) Find the domain and range of (b) Find the zero(s) of (c) Determine the intervals over which is increasing, decreasing, or constant. (d) Approximate any relative minimum or relative maximum values of (e) Is even, odd, or neither?

In Exercises 93 and 94, determine whether the statement is true or false. Justify your answer. A function with a square root cannot have a domain that is the set of real numbers

In Exercises 93 and 94, determine whether the statement is true or false. Justify your answer. It is possible for an odd function to have the interval as its domain.

In Exercises 95 and 96, find the coordinates of a second point on the graph of a function when the given point is on the graph and the function is (a) even and (b) odd. 2a, 2c53, 7f

In Exercises 95 and 96, find the coordinates of a second point on the graph of a function when the given point is on the graph and the function is (a) even and (b) odd. 2a, 2c5

Graphical Reasoning Graph each of the functions with a graphing utility. Determine whether the function is even, odd, or neither. What do you notice about the equations of functions that are odd? What do you notice about the equations of functions that are even? Can you describe a way to identify a function as odd or even by inspecting the equation? Can you describe a way to identify a function as neither odd nor even by inspecting the equation?

Even, Odd, or Neither? If is an even function, determine whether is even, odd, or neither. Explain. (a) (b) (c) gxfx2 (d) gxfx 2 gx fxgxfx g f pxx9 3x5 x kxx

In Exercises 19, match each function with its name. fxxfxx

In Exercises 19, match each function with its name. fxx fx

In Exercises 19, match each function with its name. fx1x

In Exercises 19, match each function with its name. fxx fx2

In Exercises 19, match each function with its name. fxx fx

In Exercises 19, match each function with its name. fxc 2

In Exercises 19, match each function with its name.fx x

In Exercises 19, match each function with its name. fxx fx3

In Exercises 19, match each function with its name. fxax b 3

Fill in the blank: The constant function and the identity function are two special types of ________ functions.

In Exercises 1114, (a) write the linear function such that it has the indicated function values and (b) sketch the graph of the function. f14, f06 f3 8,

In Exercises 1114, (a) write the linear function such that it has the indicated function values and (b) sketch the graph of the function. f3 8, f12 f 978

In Exercises 1114, (a) write the linear function such that it has the indicated function values and (b) sketch the graph of the function. f5 1, f5 1 f14, f0

In Exercises 1114, (a) write the linear function such that it has the indicated function values and (b) sketch the graph of the function. ff4 11 2 3 1

In Exercises 1526, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. fx2.5x 4.25 x

In Exercises 1526, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window.fx5 6 2 3 fx2

In Exercises 1526, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. fx2.5x 4.25 x

In Exercises 1526, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. fx3x gx 2x 2 1.75 2 fx5

In Exercises 1526, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. fxx 3 2 3 1 fx3

In Exercises 1526, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. fxx 1fxx 3 2 3 1 fx3

In Exercises 1526, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. fx4 2x hx

In Exercises 1526, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window.hxx 2 3 f

In Exercises 1526, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. fx4 1xkx1x

In Exercises 1526, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. kx1x 3 fx4

In Exercises 1526, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. gx x 5 fx4

In Exercises 1526, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. fx x 1 gx x

In Exercises 2730, evaluate the function for the indicated values. fxx 2 f2.1f2.9f3.1 fxx

In Exercises 2730, evaluate the function for the indicated values. hxx 3 f (a) (b) (c) (d) 2 h2 hxhh4.2h21.61 2 h2

In Exercises 2730, evaluate the function for the indicated values. kx12x 6 k5k6.1k0.1k15kx1

In Exercises 2730, evaluate the function for the indicated values. gx 7x 46 k5k 2 gg9g41 8 gx

In Exercises 3134, sketch the graph of the function. gx xgx4

In Exercises 3134, sketch the graph of the function. gx4 x

In Exercises 3134, sketch the graph of the function. gxx1 gxx

In Exercises 3134, sketch the graph of the function. gxx 3g

In Exercises 3540, sketch the graph of the function. gx x 6, 1 2x 4, x 4 x > 4 gxx

In Exercises 3540, sketch the graph of the function. fx 4 x, 4 x, x < 0 x 0 gx

In Exercises 3540, sketch the graph of the function. x 1 x 12, x 2, x 2 x > 2 fx 

In Exercises 3540, sketch the graph of the function. fx x 2 5, x 2 4x 3, x 1 x > 1 fx

In Exercises 3540, sketch the graph of the function. hx 4 x2, 3 x, x2 1, x < 2 2 x < 0 x 0 fx

In Exercises 3540, sketch the graph of the function. kx 2x 1, 2x2 1, 1 x2, x 1 1 < x 1 x > 1 hx

In Exercises 41 and 42, (a) use a graphing utility to graph the function and (b) state the domain and range of the function. x2 1 4x  1 4x kx

In Exercises 41 and 42, (a) use a graphing utility to graph the function and (b) state the domain and range of the function. kx4 1 2x  1 2x 2 sx2

Wages A mechanics pay is $14.00 per hour for regular time and time-and-a-half for overtime. The weekly wage function is where is the number of hours worked in a week. (a) Evaluate and (b) The company increased the regular work week to 45 hours. What is the new weekly wage function?

Revenue The table shows the monthly revenue (in thousands of dollars) of a landscaping business for each month of the year 2013, with representing January. A mathematical model that represents these data is (a) Use a graphing utility to graph the model. What is the domain of each part of the piecewise-defined function? How can you tell? Explain your reasoning. (b) Find and and interpret your results in the context of the problem. (c) How do the values obtained from the model in part (a) compare with the actual data values?

Fluid Flow The intake pipe of a 100-gallon tank has a flow rate of 10 gallons per minute, and two drainpipes have flow rates of 5 gallons per minute each. The figure shows the volume of fluid in the tank as a function of time Determine the combination of the input pipe and drain pipes in which the fluid is flowing in specific subintervals of the 1 hour of time shown on the graph. (There are many correct answers.)

Delivery Charges The cost of sending an overnight package from New York to Atlanta is $26.10 for a package weighing up to, but not including, 1 pound and $4.35 for each additional pound or portion of a pound. (a) Use the greatest integer function to create a model for the cost of overnight delivery of a package weighing pounds, (b) Sketch the graph of the function.

During a nine-hour snowstorm, it snows at a rate of 1 inch per hour for the first 2 hours, at a rate of 2 inches per hour for the next 6 hours, and at a rate of 0.5 inch per hour for the final hour. Write and graph a piecewise-defined function that gives the depth of the snow during the snowstorm. How many inches of snow accumulated from the storm?

HOW DO YOU SEE IT? For each graph of shown below, answer (a)(d). (a) Find the domain and range of (b) Find the - and -intercepts of the graph of (c) Determine the intervals on which is increasing, decreasing, or constant. (d) Determine whether is even, odd, or neither. Then describe the symmetry

In Exercises 49 and 50, determine whether the statement is true or false. Justify your answer. A piecewise-defined function will always have at least one -intercept or at least one -intercept.

In Exercises 49 and 50, determine whether the statement is true or false. Justify your answer. A linear equation will always have an -intercept and a y-intercept.

Horizontal shifts, vertical shifts, and reflections are called ________ transformations.

. A reflection in the -axis of is represented by ________, while a reflection in the -axis of is represented by y fxhx x y f

A nonrigid transformation of represented by is a ________ ________ when and a ________ ________ when

Match the rigid transformation of with the correct representation of the graph of where (a) (i) A horizontal shift of units to the right (b) (ii) A vertical shift of units down (c) (iii) A horizontal shift of units to the left (d) (iv) A vertical shift of units up

Shifts in the Graph of a Function For each function, sketch (on the same set of coordinate axes) a graph of each function for 1, and 3. fxx cfxxc c 1,

Shifts in the Graph of a Function For each function, sketch (on the same set of coordinate axes) a graph of each function for 1, and 3. fxx c fxx c c 3,

Shifts in the Graph of a Function For each function, sketch (on the same set of coordinate axes) a graph of each function for 0, and 2. fxxc fxx c c 2,

Shifts in the Graph of a Function For each function, sketch (on the same set of coordinate axes) a graph of each function for 1, and 3. xx2 c, x2 c, x < 0 x 0 c fxx c2 , x c2, x < 0 x 0 fxx2

In Exercises 9 and 10, use the graph of to sketch each graph. To print an enlarged copy of the graph, go to MathGraphs.com y f2x y fx1 y fx3 y fx 4 y 2 fx y fx4 x 4 4 (2, 2) (4, 2) (6, 2) 8 8 f y (0, 2) y fx f fxx c2 , x

In Exercises 9 and 10, use the graph of to sketch each graph. To print an enlarged copy of the graph, go to MathGraphs.com y f 1 3x y fx10 y fx y fx 1 y 1 3 fx y fx3 x 2 6 2 6 (0, 5) (3, 0) 10 6 14 10 f y (6, 4) (3, 0) (6, 4) y fx 5 y f2x y fx1

Writing Equations from Graphs Use the graph of to write an equation for each function whose graph is shown.

Writing Equations from Graphs Use the graph of to write an equation for each function whose graph is shown.

Writing Equations from Graphs Use the graph of to write an equation for each function whose graph is shown

Writing Equations from Graphs Use the graph of to write an equation for each function whose graph is shown.

In Exercises 1520, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph.

In Exercises 1520, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph.

In Exercises 1520, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph.

In Exercises 1520, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph.

In Exercises 1520, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph.

In Exercises 1520, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph.

In Exercises 2146, is related to one of the parent functions described in Section 2.4. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f gx12 x2 f.

In Exercises 2146, is related to one of the parent functions described in Section 2.4. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f gxx 82 gx12

In Exercises 2146, is related to one of the parent functions described in Section 2.4. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f gxx 3 1 3 7 gx

In Exercises 2146, is related to one of the parent functions described in Section 2.4. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f gx x gxx 3 1 3 7 g

In Exercises 2146, is related to one of the parent functions described in Section 2.4. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f gx2 3x2 4 g

In Exercises 2146, is related to one of the parent functions described in Section 2.4. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f gx2x 72 gx2

In Exercises 2146, is related to one of the parent functions described in Section 2.4. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f gx2 x 52 2 2 gx2x

In Exercises 2146, is related to one of the parent functions described in Section 2.4. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f gx 1 4x 2gx2 x 52 2 2 gx2x 72 g

In Exercises 2146, is related to one of the parent functions described in Section 2.4. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f gx3x x gx1 4 gx3x x gx 1

In Exercises 2146, is related to one of the parent functions described in Section 2.4. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f gx1 4 gx3x x gx 1

In Exercises 2146, is related to one of the parent functions described in Section 2.4. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f gxx 13 10 3 2 gx1

In Exercises 2146, is related to one of the parent functions described in Section 2.4. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f xx 3gxx 13 10 3 2 gx1

In Exercises 2146, is related to one of the parent functions described in Section 2.4. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f gx3x 2)3 gxx

In Exercises 2146, is related to one of the parent functions described in Section 2.4. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f gx 1 2x 13 gx3x

In Exercises 2146, is related to one of the parent functions described in Section 2.4. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f g x 5x x2 gx

In Exercises 2146, is related to one of the parent functions described in Section 2.4. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f gx6 g x 5x x

In Exercises 2146, is related to one of the parent functions described in Section 2.4. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f gx x 48 gx

In Exercises 2146, is related to one of the parent functions described in Section 2.4. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f gxx 39 gx

In Exercises 2146, is related to one of the parent functions described in Section 2.4. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f x 2x 14 x 23

In Exercises 2146, is related to one of the parent functions described in Section 2.4. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f x1 2gx 2x 14 x 23 gx x 4

In Exercises 2146, is related to one of the parent functions described in Section 2.4. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f gx3 xgx2

In Exercises 2146, is related to one of the parent functions described in Section 2.4. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f gx2x 5 g

In Exercises 2146, is related to one of the parent functions described in Section 2.4. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f gxx 9 gxx

In Exercises 2146, is related to one of the parent functions described in Section 2.4. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f gxx 4 8 g

In Exercises 2146, is related to one of the parent functions described in Section 2.4. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f gx7 x 2 gx3

In Exercises 2146, is related to one of the parent functions described in Section 2.4. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f gx3x 1 g

Exercises 4754, write an equation for the function described by the given characteristics. The shape of , but shifted three units to the right and seven units down

Exercises 4754, write an equation for the function described by the given characteristics. The shape of , but shifted two units to the left, nine units up, and then reflected in the -axis

Exercises 4754, write an equation for the function described by the given characteristics. The shape of , but shifted 13 units to the right

Exercises 4754, write an equation for the function described by the given characteristics. The shape of , but shifted six units to the left, six units down, and then reflected in the -axis

Exercises 4754, write an equation for the function described by the given characteristics. The shape of but shifted 12 units up and then reflected in the -axis

Exercises 4754, write an equation for the function described by the given characteristics. The shape of but shifted four units to the left and eight units down

Exercises 4754, write an equation for the function described by the given characteristics. The shape of but shifted six units to the left and then reflected in both the -axis and the -axis

Exercises 4754, write an equation for the function described by the given characteristics. The shape of but shifted nine units down and then reflected in both the -axis and the -axis

Writing Equations from Graphs Use the graph of to write an equation for each function whose graph is shown

Writing Equations from Graphs Use the graph of to write an equation for each function whose graph is shown.

Writing Equations from Graphs Use the graph of to write an equation for each function whose graph is shown.

Writing Equations from Graphs Use the graph of to write an equation for each function whose graph is shown.

In Exercises 5964, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph. Then use a graphing utility to verify your answer.

In Exercises 5964, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph. Then use a graphing utility to verify your answer.

In Exercises 5964, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph. Then use a graphing utility to verify your answer.

In Exercises 5964, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph. Then use a graphing utility to verify your answer.

In Exercises 5964, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph. Then use a graphing utility to verify your answer.

In Exercises 5964, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph. Then use a graphing utility to verify your answer.

Graphical Analysis In Exercises 6568, use the viewing window shown to write a possible equation for the transformation of the parent function.

Graphical Analysis In Exercises 6568, use the viewing window shown to write a possible equation for the transformation of the parent function.

Graphical Analysis In Exercises 6568, use the viewing window shown to write a possible equation for the transformation of the parent function.

Graphical Analysis In Exercises 6568, use the viewing window shown to write a possible equation for the transformation of the parent function.

The number of horsepower required to overcome wind drag on an automobile is approximated by where is the speed of the car (in miles per hour). (a) Use a graphing utility to graph the function. (b) Rewrite the horsepower function so that represents the speed in kilometers per hour. [Find ] Identify the type of transformation applied to the graph of the horsepower function

Households The numbers (in millions) of households in the United States from 2003 through 2010 can be approximated by where represents the year, with corresponding to 2003. (Source: U.S. Census Bureau) (a) Describe the transformation of the parent function Then use a graphing utility to graph the function over the specified domain. (b) Find the average rate of change of the function from 2003 to 2010. Interpret your answer in the context of the problem. (c) Use the model to predict the number of households in the United States in 2018. Does your answer seem reasonable? Explain.

In Exercises 7174, determine whether the statement is true or false. Justify your answer. The graph of is a reflection of the graph of in the -axis.

In Exercises 7174, determine whether the statement is true or false. Justify your answer. . The graph of is a reflection of the graph of in the -axis.

In Exercises 7174, determine whether the statement is true or false. Justify your answer. The graphs of and are identical.

In Exercises 7174, determine whether the statement is true or false. Justify your answer.If the graph of the parent function is shifted six units to the right, three units up, and reflected in the -axis, then the point will lie on the graph of the transformation

Finding Points on a Graph The graph of passes through the points and Find the corresponding points on the graph of

Think About It You can use either of two methods to graph a function: plotting points or translating a parent function as shown in this section. Which method of graphing do you prefer to use for each function? Explain. fx3x2 4x 1 y f fx2x 12 6 fx3x2

Predicting Graphical Relationships Use a graphing utility to graph and in the same viewing window. Before looking at the graphs, try to predict how the graphs of and relate to the graph of f fxx 2, gxx 42, hg f. hxx 42 3 fxx 2fxx 2, g gxx 12, hxx hxx 12 2 fxx 2fxx 2, g gxx 42, hxx hxx 42 2 fxx

HOW DO YOU SEE IT? Use the graph of to find the intervals on which each of the graphs in (a)(e) is increasing and decreasing. If not possible, then state the reason. (a) (b) (c) (d) y fx 1(e) y fx 21 y 1 2 y fxy fxfx x y 4 24 2

Describing Profits Management originally predicted that the profits from the sales of a new product would be approximated by the graph of the function shown. The actual profits are shown by the function along with a verbal description. Use the concepts of transformations of graphs to write in terms of (a) The profits were only three-fourths as large as expected. (b) The profits were consistently $10,000 greater than predicted. (c) There was a two-year delay in the introduction of the product. After sales began, profits grew as expected.

Reversing the Order of Transformations Reverse the order of transformations in Example 2(a). Do you obtain the same graph? Do the same for Example 2(b). Do you obtain the same graph? Explain.

Two functions and can be combined by the arithmetic operations of ________, ________, ________, and _________ to create new functions.

The ________ of the function with is f gf gxfgx. gf Gra

In Exercises 3 and 4, use the graphs of and to graph To print an enlarged copy of the graph, go to MathGraphs.com

In Exercises 3 and 4, use the graphs of and to graph To print an enlarged copy of the graph, go to MathGraphs.com

In Exercises 512, find (a) (b) (c) and (d) What is the domain of fxx 2, gxx 2f fg

In Exercises 512, find (a) (b) (c) and (d) What is the domain of fx2x 5, gx2 xfxx 

In Exercises 512, find (a) (b) (c) and (d) What is the domain of fxx gx4x 5 2,fx2x

In Exercises 512, find (a) (b) (c) and (d) What is the domain of fx3x 1, gx5x 4fxx g

In Exercises 512, find (a) (b) (c) and (d) What is the domain of fxx gx1 x 2 6,fx3x

In Exercises 512, find (a) (b) (c) and (d) What is the domain of x2 1 fxx2 4,fx

In Exercises 512, find (a) (b) (c) and (d) What is the domain of x2 fx1x

In Exercises 512, find (a) (b) (c) and (d) What is the domain of gxx3 fxxx 1

In Exercises 1324, evaluate the indicated function for and f g2f g

In Exercises 1324, evaluate the indicated function for and f g1fx

In Exercises 1324, evaluate the indicated function for and f g0f g

In Exercises 1324, evaluate the indicated function for and f g1f 

In Exercises 1324, evaluate the indicated function for and f g3tf 

In Exercises 1324, evaluate the indicated function for and f gt 2f

In Exercises 1324, evaluate the indicated function for and fg6f

In Exercises 1324, evaluate the indicated function for and fg6fg6

In Exercises 1324, evaluate the indicated function for and fg5fg

In Exercises 1324, evaluate the indicated function for and fg0fg

In Exercises 1324, evaluate the indicated function for and fg1g3fg0

In Exercises 1324, evaluate the indicated function for and fg5f4fg1

In Exercises 25 and 26, graph the functions and on the same set of coordinate axes. fxgxx 1 12 x,

In Exercises 25 and 26, graph the functions and on the same set of coordinate axes. fx4 x gxx 2,fxgxx

In Exercises 2730, use a graphing utility to graph and in the same viewing window. Which function contributes most to the magnitude of the sum when Which function contributes most to the magnitude of the sum when 10 fx3x,x

In Exercises 2730, use a graphing utility to graph and in the same viewing window. Which function contributes most to the magnitude of the sum when Which function contributes most to the magnitude of the sum when fxgxxx2

In Exercises 2730, use a graphing utility to graph and in the same viewing window. Which function contributes most to the magnitude of the sum when Which function contributes most to the magnitude of the sum when fx3x 2, gx x 5fxgx

In Exercises 2730, use a graphing utility to graph and in the same viewing window. Which function contributes most to the magnitude of the sum when Which function contributes most to the magnitude of the sum when gx 3x fxx 2 1 2 12,f

In Exercises 3134, find (a) (b) and (c fxx gxx 1 2,g g 

In Exercises 3134, find (a) (b) and (c fx3x 5, gx5 xfxx g

In Exercises 3134, find (a) (b) and (c gxx fx3 1 3 x 

In Exercises 3134, find (a) (b) and (c fxx3, gx1

In Exercises 3542, find (a) and (b) Find the domain of each function and each composite function. gxx 2 fxx 4,g f.

In Exercises 3542, find (a) and (b) Find the domain of each function and each composite function. gxx fx3 1 3 x 

In Exercises 3542, find (a) and (b) Find the domain of each function and each composite function. fxx gxx 2 1,gxx f

In Exercises 3542, find (a) and (b) Find the domain of each function and each composite function. gxx6 fxx23,fxx g

In Exercises 3542, find (a) and (b) Find the domain of each function and each composite function. fxx, gxx 6gxx6

In Exercises 3542, find (a) and (b) Find the domain of each function and each composite function. fxgx3 x x 4,fxx

In Exercises 3542, find (a) and (b) Find the domain of each function and each composite function. fxgxx 3 1x,

In Exercises 3542, find (a) and (b) Find the domain of each function and each composite function. fxgxx 1 3x2 1

In Exercises 4346, use the graphs of and to evaluate the functions. f g3fg2x1 23

In Exercises 4346, use the graphs of and to evaluate the functions. f g1fg4f g3f

In Exercises 4346, use the graphs of and to evaluate the functions. g f f 2g2f g1fg

In Exercises 4346, use the graphs of and to evaluate the functions. g f f3g1g f f

In Exercises 4754, find two functions and such that (There are many correct answers.) hx2x 12f

In Exercises 4754, find two functions and such that (There are many correct answers.) hx1 x3 hx2x

In Exercises 4754, find two functions and such that (There are many correct answers.) hxhx9 x 3 x 2 4hx1

In Exercises 4754, find two functions and such that (There are many correct answers.) hx9 x 

In Exercises 4754, find two functions and such that (There are many correct answers.) hx1x 2h

In Exercises 4754, find two functions and such that (There are many correct answers.) hx45x 22 hx1

In Exercises 4754, find two functions and such that (There are many correct answers.) hxx2 34 x2hx

In Exercises 4754, find two functions and such that (There are many correct answers.) hx27x 3 6x10 27x 3h

Stopping Distance The research and development department of an automobile manufacturer has determined that when a driver is required to stop quickly to avoid an accident, the distance (in feet) the car travels during the drivers reaction time is given by where is the speed of the car in miles per hour. The distance (in feet) traveled while the driver is braking is given by (a) Find the function that represents the total stopping distance (b) Graph the functions and on the same set of coordinate axes for (c) Which function contributes most to the magnitude of the sum at higher speeds? Explain.

Vital Statistics Let be the number of births in the United States in year and let represent the number of deaths in the United States in year where corresponds to 2010. (a) If is the population of the United States in year then find the function that represents the percent change in the population of the United States. (b) Interpret the value of

Let be the number of dogs in the United States in year and let be the number of cats in the United States in year where corresponds to 2010. (a) Find the function that represents the total number of dogs and cats in the United States. (b) Interpret the value of (c) Let represent the population of the United States in year where corresponds to 2010. Find and interpret htpt nt . t

Graphical Reasoning An electronically controlled thermostat in a home lowers the temperature automatically during the night. The temperature in the house (in degrees Fahrenheit) is given in terms of the time in hours on a 24-hour clock (see figure). (a) Explain why is a function of (b) Approximate and (c) The thermostat is reprogrammed to produce a temperature for which How does this change the temperature? (d) The thermostat is reprogrammed to produce a temperature for which How does this change the temperature? (e) Write a piecewise-defined function that represents the graph.

Geometry A square concrete foundation is a base for a cylindrical tank (see figure). (a) Write the radius of the tank as a function of the length of the sides of the square. (b) Write the area of the circular base of the tank as a function of the radius (c) Find and interpret A rx. r.

Bacteria Count The number of bacteria in a refrigerated food is given by where is the temperature of the food in degrees Celsius. When the food is removed from refrigeration, the temperature of the food is given by where is the time in hours. (a) Find the composition and interpret its meaning in context. (b) Find the bacteria count after 0.5 hour.

Salary You are a sales representative for a clothing manufacturer. You are paid an annual salary, plus a bonus of 3% of your sales over $500,000. Consider the two functions and When is greater than $500,000, which of the following represents your bonus? Explain your reasoning. (a) fgxx (b) gfxfgx

Consumer Awareness The suggested retail price of a new hybrid car is dollars. The dealership advertises a factory rebate of $2000 and a 10% discount. (a) Write a function in terms of giving the cost of the hybrid car after receiving the rebate from the factory. (b) Write a function in terms of giving the cost of the hybrid car after receiving the dealership discount. (c) Form the composite functions and and interpret each. (d) Find and Which yields the lower cost for the hybrid car? Explain.

In Exercises 63 and 64, three siblings are three different ages. The oldest is twice the age of the middle sibling, and the middle sibling is six years older than one-half the age of the youngest. (a) Write a composite function that gives the oldest siblings age in terms of the youngest. Explain how you arrived at your answer. (b) If the oldest sibling is 16 years old, then find the ages of the other two siblings

In Exercises 63 and 64, three siblings are three different ages. The oldest is twice the age of the middle sibling, and the middle sibling is six years older than one-half the age of the youngest. (a) Write a composite function that gives the youngest siblings age in terms of the oldest. Explain how you arrived at your answer. (b) If the youngest sibling is two years old, then findthe ages of the other two siblings.

In Exercises 65 and 66, determine whether the statement is true or false. Justify your answer fxx 1 gx6x,R S  f g)xg f)x.fxx

In Exercises 65 and 66, determine whether the statement is true or false. Justify your answer When you are given two functions and , you can calculate if and only if the range of is asubset of the domain of f

Proof Prove that the product of two odd functions is an even function, and that the product of two even functions is an even function.

HOW DO YOU SEE IT? The graphs labeled and represent four different pricing discounts, where is the original price (in dollars) and is the sale price (in dollars). Match each function with its graph. Describe the situations in parts (c) and (d). (a) A 50% discount is applied. (b) A $5 discount is applied. (c) (d) f gp g fp gp: f

Conjecture Use examples to hypothesize whether the product of an odd function and an even function is even or odd. Then prove your hypothesis.

Proof (a) Given a function prove that is even and is odd, where and (b) Use the result of part (a) to prove that any function can be written as a sum of even and odd functions. [Hint: Add the two equations in part (a).] (c) Use the result of part (b) to write each function as a sum of even and odd functions. fxx2 2x 1, kx1 x 1 hx1

If the composite functions and both equal then the function is the ________ function of f

The inverse function of is denoted by ________.

The domain of is the ________ of and the ________ of is the range of f

The graphs of and are reflections of each other in the line ________.

A function is ________ when each value of the dependent variable corresponds to exactly one value of the independent variable.

A graphical test for the existence of an inverse function of is called the _______ Line Test

In Exercises 712, find the inverse function of informally. Verify that f1 f f fx fx6x x

In Exercises 712, find the inverse function of informally. Verify that f1 f f fx fx13 fx

In Exercises 712, find the inverse function of informally. Verify that f1 f f fx fx3x 1fx

In Exercises 712, find the inverse function of informally. Verify that f1 f f fx fxx 15 fx3

In Exercises 712, find the inverse function of informally. Verify that f1 f f fx fx3 xf

In Exercises 712, find the inverse function of informally. Verify that f1 f f fx fxx5 fx

In Exercises 1316, verify that and are inverse functions. 7 fx 72

In Exercises 1316, verify that and are inverse functions. fxgx4x 9 x 94 ,g

In Exercises 1316, verify that and are inverse functions. 3 fxx x 5 3 5,fxg

In Exercises 1316, verify that and are inverse functions. 3 fx2xx3

In Exercises 1720, use the graph of the function to sketch the graph of its inverse function y f 1

In Exercises 1720, use the graph of the function to sketch the graph of its inverse function y f 2

In Exercises 1720, use the graph of the function to sketch the graph of its inverse function y f 3

In Exercises 1720, use the graph of the function to sketch the graph of its inverse function y f 4

In Exercises 2132, verify that and are inverse functions (a) algebraically and (b) graphically. 2 fx2x,gf

In Exercises 2132, verify that and are inverse functions (a) algebraically and (b) graphically. fxx 5, gxx 5gxx2

In Exercises 2132, verify that and are inverse functions (a) algebraically and (b) graphically. 7 fx7x 1,fx

In Exercises 2132, verify that and are inverse functions (a) algebraically and (b) graphically. 4 fx3 4x,gxx

In Exercises 2132, verify that and are inverse functions (a) algebraically and (b) graphically. 3 fx8xx3

In Exercises 2132, verify that and are inverse functions (a) algebraically and (b) graphically. fx1x gx1x

In Exercises 2132, verify that and are inverse functions (a) algebraically and (b) graphically. gxx x 0 fxx 4, 2 4,gx1x

In Exercises 2132, verify that and are inverse functions (a) algebraically and (b) graphically. 3 fx1 x 1 x 3,gxx

In Exercises 2132, verify that and are inverse functions (a) algebraically and (b) graphically. fx9 x x 0, gx9 x, x 9 2,gx3

In Exercises 2132, verify that and are inverse functions (a) algebraically and (b) graphically. fxx 0, , 11

In Exercises 2132, verify that and are inverse functions (a) algebraically and (b) graphically. x 1 fxx 1x 5

In Exercises 2132, verify that and are inverse functions (a) algebraically and (b) graphically. x 1 fxx 3x 2

In Exercises 33 and 34, does the function have an inverse function?

In Exercises 33 and 34, does the function have an inverse function?

In Exercises 35 and 36, use the table of values for to complete a table for

In Exercises 35 and 36, use the table of values for to complete a table for

In Exercises 3740, does the function have an inverse function?

In Exercises 3740, does the function have an inverse function?

In Exercises 3740, does the function have an inverse function?

In Exercises 3740, does the function have an inverse function?

n Exercises 4144, use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function has an inverse function gxx 52 1 3

n Exercises 4144, use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function has an inverse function fx18x 2gxx

n Exercises 4144, use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function has an inverse function fx 2x16 x2fx1

n Exercises 4144, use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function has an inverse function hxx 4x 4fx

In Exercises 4556, (a) find the inverse function of (b) graph both and on the same set of coordinate axes, (c) describe the relationship between the graphs of and and (d) state the domains and ranges of fx2x 3 fx3

In Exercises 4556, (a) find the inverse function of (b) graph both and on the same set of coordinate axes, (c) describe the relationship between the graphs of and and (d) state the domains and ranges of fx3x 1f

In Exercises 4556, (a) find the inverse function of (b) graph both and on the same set of coordinate axes, (c) describe the relationship between the graphs of and and (d) state the domains and ranges of fxx 3 1 5 2fx

In Exercises 4556, (a) find the inverse function of (b) graph both and on the same set of coordinate axes, (c) describe the relationship between the graphs of and and (d) state the domains and ranges of fxx fxx 3 1 5 f

In Exercises 4556, (a) find the inverse function of (b) graph both and on the same set of coordinate axes, (c) describe the relationship between the graphs of and and (d) state the domains and ranges of fx4 x 0 x 2 2

In Exercises 4556, (a) find the inverse function of (b) graph both and on the same set of coordinate axes, (c) describe the relationship between the graphs of and and (d) state the domains and ranges of fxx x 0 2 2,fx

In Exercises 4556, (a) find the inverse function of (b) graph both and on the same set of coordinate axes, (c) describe the relationship between the graphs of and and (d) state the domains and ranges of fx4xf

In Exercises 4556, (a) find the inverse function of (b) graph both and on the same set of coordinate axes, (c) describe the relationship between the graphs of and and (d) state the domains and ranges of fx 2x fx4

In Exercises 4556, (a) find the inverse function of (b) graph both and on the same set of coordinate axes, (c) describe the relationship between the graphs of and and (d) state the domains and ranges of fxx 1x 2fx

In Exercises 4556, (a) find the inverse function of (b) graph both and on the same set of coordinate axes, (c) describe the relationship between the graphs of and and (d) state the domains and ranges of fxx 3x 2 fxx

In Exercises 4556, (a) find the inverse function of (b) graph both and on the same set of coordinate axes, (c) describe the relationship between the graphs of and and (d) state the domains and ranges of fx3 x 1fxx

In Exercises 4556, (a) find the inverse function of (b) graph both and on the same set of coordinate axes, (c) describe the relationship between the graphs of and and (d) state the domains and ranges of xx35 fx

In Exercises 5772, determine whether the function has an inverse function. If it does, then find the inverse function. fxx4f

In Exercises 5772, determine whether the function has an inverse function. If it does, then find the inverse function. fx1x 2 fx

In Exercises 5772, determine whether the function has an inverse function. If it does, then find the inverse function. gxfx3x 5x8fx1

In Exercises 5772, determine whether the function has an inverse function. If it does, then find the inverse function. fx3x 5x

In Exercises 5772, determine whether the function has an inverse function. If it does, then find the inverse function. px 4gx

In Exercises 5772, determine whether the function has an inverse function. If it does, then find the inverse function. fx3x 45 px

In Exercises 5772, determine whether the function has an inverse function. If it does, then find the inverse function. fxx 3x 3 2,f

In Exercises 5772, determine whether the function has an inverse function. If it does, then find the inverse function. qxx 52fxx

In Exercises 5772, determine whether the function has an inverse function. If it does, then find the inverse function. fx x 3,6 x,x < 0x 0qx

In Exercises 5772, determine whether the function has an inverse function. If it does, then find the inverse function. fx x,x2 3x, x 0 x > 0fx

In Exercises 5772, determine whether the function has an inverse function. If it does, then find the inverse function. hx 4x2fx

In Exercises 5772, determine whether the function has an inverse function. If it does, then find the inverse function. fxx 2, x 2hx

In Exercises 5772, determine whether the function has an inverse function. If it does, then find the inverse function. fx2x 3fx

In Exercises 5772, determine whether the function has an inverse function. If it does, then find the inverse function. fxx 2fx

In Exercises 5772, determine whether the function has an inverse function. If it does, then find the inverse function. fx6x 44x

In Exercises 5772, determine whether the function has an inverse function. If it does, then find the inverse function. fx5x 32x

In Exercises 7382, restrict the domain of the function so that the function is one-to-one and has an inverse function. Then find the inverse function State the domains and ranges of and Explain your results. (There are many correct answers.) fxx 22 f1.

In Exercises 7382, restrict the domain of the function so that the function is one-to-one and has an inverse function. Then find the inverse function State the domains and ranges of and Explain your results. (There are many correct answers.) fx1 x 4 fxx

In Exercises 7382, restrict the domain of the function so that the function is one-to-one and has an inverse function. Then find the inverse function State the domains and ranges of and Explain your results. (There are many correct answers.) fxx 2 f

In Exercises 7382, restrict the domain of the function so that the function is one-to-one and has an inverse function. Then find the inverse function State the domains and ranges of and Explain your results. (There are many correct answers.) fxx 5fxx

In Exercises 7382, restrict the domain of the function so that the function is one-to-one and has an inverse function. Then find the inverse function State the domains and ranges of and Explain your results. (There are many correct answers.) fxx 62 fx

In Exercises 7382, restrict the domain of the function so that the function is one-to-one and has an inverse function. Then find the inverse function State the domains and ranges of and Explain your results. (There are many correct answers.) fxx 42 fxx 

In Exercises 7382, restrict the domain of the function so that the function is one-to-one and has an inverse function. Then find the inverse function State the domains and ranges of and Explain your results. (There are many correct answers.) fx 2x 2 1 2 5 fxx

In Exercises 7382, restrict the domain of the function so that the function is one-to-one and has an inverse function. Then find the inverse function State the domains and ranges of and Explain your results. (There are many correct answers.) fx1 2 x fx 2x 2 1 2 5 f

In Exercises 7382, restrict the domain of the function so that the function is one-to-one and has an inverse function. Then find the inverse function State the domains and ranges of and Explain your results. (There are many correct answers.) fxx 41 x 1

In Exercises 7382, restrict the domain of the function so that the function is one-to-one and has an inverse function. Then find the inverse function State the domains and ranges of and Explain your results. (There are many correct answers.) fx fxx 41 x 12 fx1 2 x

In Exercises 8388, use the functions and to find the indicated value or function. f 31 g 11 g x x3

In Exercises 8388, use the functions and to find the indicated value or function. g1 f 1 f 31 g 1

In Exercises 8388, use the functions and to find the indicated value or function. f 41 f 1 6 g1 f 1

In Exercises 8388, use the functions and to find the indicated value or function. g1 g1 f 41 f

In Exercises 8388, use the functions and to find the indicated value or function. f g1 g

In Exercises 8388, use the functions and to find the indicated value or function. g1 f 1 f

In Exercises 8992, use the functions and to find the specified function. g1 f 1 f

In Exercises 8992, use the functions and to find the specified function. f 1 g1 g

In Exercises 8992, use the functions and to find the specified function. f g1 f

In Exercises 8992, use the functions and to find the specified function. g f 1 f

Hourly Wage Your wage is $10.00 per hour plus $0.75 for each unit produced per hour. So, your hourly wage in terms of the number of units produced is (a) Find the inverse function. What does each variable represent in the inverse function? (b) Determine the number of units produced when your hourly wage is $24.25.

The function approximates the exhaust temperature in degrees Fahrenheit, where is the percent load for a diesel engine. (a) Find the inverse function. What does each variable represent in the inverse function? (b) Use a graphing utility to graph the inverse function. (c) The exhaust temperature of the engine must not exceed 500 degrees Fahrenheit. What is the percent load interval?

In Exercises 95 and 96, determine whether the statement is true or false. Justify your answer. If is an even function, then exists

In Exercises 95 and 96, determine whether the statement is true or false. Justify your answer. If the inverse function of exists and the graph of has a -intercept, then the -intercept of is an -intercept of

In Exercises 97 and 98, use the graph of the function to create a table of values for the given points. Then create a second table that can be used to find and sketch the graph of if possible.

In Exercises 97 and 98, use the graph of the function to create a table of values for the given points. Then create a second table that can be used to find and sketch the graph of if possible.

Proof Prove that if and are one-to-one functions, then f g1xg1 f 1x.gfx2

Proof Prove that if is a one-to-one odd function, then is an odd function.

Think About It The function has an inverse function, and Find

Think About It Consider the functions and Evaluate and for the indicated values of What can you conclude about the functions?

Think About It Restrict the domain of to Use a graphing utility to graph the function. Does the restricted function have an inverse function? Explain.

HOW DO YOU SEE IT? The cost for a business to make personalized T-shirts is given by where represents the number of T-shirts. (a) The graphs of and are shown below. Match each function with its graph. (b) Explain what and represent in the context of the problem. C1 Cxx

In Exercises 105 and 106, determine whether the situation could be represented by a one-to-one function. If so, then write a statement that best describes the inverse function. The number of miles a marathon runner has completedin terms of the time in hours

In Exercises 105 and 106, determine whether the situation could be represented by a one-to-one function. If so, then write a statement that best describes the inverse function. The depth of the tide at a beach in terms of the time over a 24-hour period

In Exercises 1 4, find the slope and -intercept (if possible) of the equation of the line. Sketch the line. y 2x 7

In Exercises 1 4, find the slope and -intercept (if possible) of the equation of the line. Sketch the line. 10x 2y 9

In Exercises 1 4, find the slope and -intercept (if possible) of the equation of the line. Sketch the line. y 6 x

In Exercises 1 4, find the slope and -intercept (if possible) of the equation of the line. Sketch the line. x 3

In Exercises 5 and 6, plot the points and find the slope of the line passing through the pair of points. 6, 4, 3, 43

In Exercises 5 and 6, plot the points and find the slope of the line passing through the pair of points. 3, 2, 8, 2

In Exercises 7 and 8, find an equation of the line that passes through the given point and has the indicated slope Sketch the line. 210, 3,m

In Exercises 7 and 8, find an equation of the line that passes through the given point and has the indicated slope Sketch the line.m 8, 5, m 0 12 1

In Exercises 9 and 10, find an equation of the line passing through the points. 1, 0, 6, 211

In Exercises 9 and 10, find an equation of the line passing through the points. 11, 2, 6, 1m

In Exercises 11 and 12, write the slope-intercept form of the equations of the lines through the given point (a) parallel to and (b) perpendicular to the given line. 5x 4y 8, 3, 22x

In Exercises 11 and 12, write the slope-intercept form of the equations of the lines through the given point (a) parallel to and (b) perpendicular to the given line. 2x 3y 5, 8, 31

Sales A discount outlet is offering a 20% discount on all items. Write a linear equation giving the sale price for an item with a list price L

Hourly Wage A microchip manufacturer pays its assembly line workers $12.25 per hour. In addition, workers receive a piecework rate of $0.75 per unit produced. Write a linear equation for the hourly wage in terms of the number of units produced per hour.

In Exercises 1518, determine whether the equation represents as a function of x 16x y 2x y 3 0 4 0

In Exercises 1518, determine whether the equation represents as a function of x 2x y 3 0

In Exercises 1518, determine whether the equation represents as a function of x y 1 x

In Exercises 1518, determine whether the equation represents as a function of x y y 1 x x 2 1

In Exercises 19 and 20, evaluate the function at each specified value of the independent variable and simplify. gxx 43 y g8gt 1g27gx gxx 43

In Exercises 19 and 20, evaluate the function at each specified value of the independent variable and simplify. hx 2x 1, x2 2, x 1 x > 1g8g h2h1h0h2 hx 2x

In Exercises 21 and 22, find the domain of the function. Verify your result with a graph. fx25 x 2h

In Exercises 21 and 22, find the domain of the function. Verify your result with a graph. h(x) x x2 x 6

In Exercises 23 and 24, the velocity of a ball projected upward from ground level is given by where is the time in seconds and is the velocity in feet per second. Find the velocity when t 1.

In Exercises 23 and 24, the velocity of a ball projected upward from ground level is given by where is the time in seconds and is the velocity in feet per second. Find the time when the ball reaches its maximum height. [Hint: Find the time when vt0.t

In Exercises 25 and 26, find the difference quotient and simplify your answer. h fx2x , h 0 2 3x 1,vt0

In Exercises 25 and 26, find the difference quotient and simplify your answer. h fxx , h 0 3 5x2 x,fx

In Exercises 27 and 28, use the Vertical Line Test to determine whether is a function of To print an enlarged copy of the graph, go to MathGraphs.com. y x 32x

In Exercises 27 and 28, use the Vertical Line Test to determine whether is a function of To print an enlarged copy of the graph, go to MathGraphs.com. x 4 yy

In Exercises 2932, find the zeros of the function algebraically fx5x2 4x 14

In Exercises 2932, find the zeros of the function algebraically fx8x 311 x fx5

In Exercises 2932, find the zeros of the function algebraically fx2x 1fx8

In Exercises 2932, find the zeros of the function algebraically fxx3 x2 fx

In Exercises 33 and 34, use a graphing utility to graph the function and visually determine the intervals on which the function is increasing, decreasing, or constant. fxxx 1fx

In Exercises 33 and 34, use a graphing utility to graph the function and visually determine the intervals on which the function is increasing, decreasing, or constant. fxx2 42fxx

In Exercises 35 and 36, use a graphing utility to graph the function and approximate (to two decimal places) any relative minima or maxima. fx x 2 1 2 2x 1978

In Exercises 35 and 36, use a graphing utility to graph the function and approximate (to two decimal places) any relative minima or maxima. fxx3 4x fx x 2 1 2 2x 1

In Exercises 37 and 38, find the average rate of change of the function from x1 0, x fx x 2 4 2 8x 4, x2 x

In Exercises 37 and 38, find the average rate of change of the function from x1 3, x fx2 x 1, 2 7 x1 0,

In Exercises 39 42, determine whether the function is even, odd, or neither. Then describe the symmetry. fxx 5 4x 7 x1

In Exercises 39 42, determine whether the function is even, odd, or neither. Then describe the symmetry. fxx 4 20x2 fx

In Exercises 39 42, determine whether the function is even, odd, or neither. Then describe the symmetry. fx2xx2 3 fx

In Exercises 39 42, determine whether the function is even, odd, or neither. Then describe the symmetry. fx 5 6x2 fx

In Exercises 43 and 44, (a) write the linear function such that it has the indicated function values, and (b) sketch the graph of the function. f2 6, f13f 2.

In Exercises 43 and 44, (a) write the linear function such that it has the indicated function values, and (b) sketch the graph of the function. f0 5, f4 8f2 6

In Exercises 4550, sketch the graph of the function. fxx2 5f0

In Exercises 4550, sketch the graph of the function. gx 3x3 f

In Exercises 4550, sketch the graph of the function. fxx 1gx

In Exercises 4550, sketch the graph of the function. gx1 x 5 fx

In Exercises 4550, sketch the graph of the function. gx x 4 g

In Exercises 4550, sketch the graph of the function. fx 5x 3, 4x 5, x 1 x < 1 gx

In Exercises 5160, is related to one of the parent functions described in this chapter. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f hxx 3 2 2 9 h

In Exercises 5160, is related to one of the parent functions described in this chapter. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f hxx 2hxx 3 2 2 9 h

In Exercises 5160, is related to one of the parent functions described in this chapter. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of fhx x 4 x 3

In Exercises 5160, is related to one of the parent functions described in this chapter. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f hxhx x 4 x 35 hxx 2

In Exercises 5160, is related to one of the parent functions described in this chapter. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f hx x 22 2 2 3 hxhx

In Exercises 5160, is related to one of the parent functions described in this chapter. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f hx1 2x 1hx x 22 2 2 3 hxhx

In Exercises 5160, is related to one of the parent functions described in this chapter. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of fhx  x 6 hx

In Exercises 5160, is related to one of the parent functions described in this chapter. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f hx x 1 9 hx1

In Exercises 5160, is related to one of the parent functions described in this chapter. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f hx5 x 9 h

In Exercises 5160, is related to one of the parent functions described in this chapter. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f hx 1 3 x 3 hx

In Exercises 61 and 62, find (a) (b) (c) and (d) What is the domain of fxx gx2x 1 2 3,f/g?

In Exercises 61 and 62, find (a) (b) (c) and (d) What is the domain of fxx gx3 x 2 4,fxx g

n Exercises 63 and 64, find (a) and (b) Find the domain of each function and each composite function fxgx3x 1 1 3 x 3

n Exercises 63 and 64, find (a) and (b) Find the domain of each function and each composite function gxx2 fxx 1, fxgx3x

In Exercises 65 and 66, the number of bacteria in a refrigerated food is given by where is the temperature of the food in degrees Celsius. When the food is removed from refrigeration, the temperature of the food is given by where is the time in hours. . Find the composition and interpret its meaning in context.

In Exercises 65 and 66, the number of bacteria in a refrigerated food is given by where is the temperature of the food in degrees Celsius. When the food is removed from refrigeration, the temperature of the food is given by where is the time in hours. Find the time when the bacteria count reaches 750.

In Exercises 67 and 68, find the inverse function of informally. Verify that and fx3 1 x 4 5 f

In Exercises 67 and 68, find the inverse function of informally. Verify that and fxx fx3 1 x 4

In Exercises 69 and 70, use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function has an inverse function. fxx 12 fxx

In Exercises 69 and 70, use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function has an inverse function. ht2 t 3 fx

In Exercises 71 and 72, (a) find the inverse function of (b) graph both and on the same set of coordinate axes, (c) describe the relationship between the graphs of and and (d) state the domains and ranges of fxfxx 1 1 2x 3 f 1 f

In Exercises 71 and 72, (a) find the inverse function of (b) graph both and on the same set of coordinate axes, (c) describe the relationship between the graphs of and and (d) state the domains and ranges of fxx 1 1

In Exercises 73 and 74, restrict the domain of the function to an interval on which the function is increasing, and determine on that interval. fx2x 42 f 1

In Exercises 73 and 74, restrict the domain of the function to an interval on which the function is increasing, and determine on that interval. fxx 2fx2

In Exercises 75 and 76, determine whether the statement is true or false. Justify your answer. Relative to the graph of the function is shifted 9 units to the left and 13 units down, then reflected in the -axis

In Exercises 75 and 76, determine whether the statement is true or false. Justify your answer. If and are two inverse functions, then the domain of g is equal to the range of f