In Exercises 14 and 15, rewrite j(x) as the composition of

In each part of the problem the graph of (x) to the left has been transformed into the graph of g(x) to the right. First describe whether the graph of (x) was stretched/compressed, reflected, and/or shifted vertically/horizontally to form g(x). Then write the equation for g(x) in terms of f(x).

Match each of the following functions with its graph. Identify the parent (original) function p(x) and the transformation(s) that took place. a. (x) (x 2)3 4 b. g(x) x3 2 c. h(x) ln(x 1) d. k(x) ex 5 5

Let (x) x3 . a. Write the equation for the new function g(x) that results from each of the following transformations of (x). Explain in words the effect of the transformations. i. (2x) iii. (x 1 2) ii. 22(x) 2 1 iv. 2(2x) b. Sketch by hand the graph of (x) and each function in part (a).

Explain in words the effect of the following transformations on the graph of g(t). a. 25g(t) c. 2g(2t) 2 4 b. g(t 2 3) 1 1 d. 4g(2t) 2 2

Decide if each graph (although not necessarily a function) is symmetric across the x-axis, across the y-axis, and/or about the origin.

Complete the partial graph shown on the next page in three different ways to create a graph that is: a. Symmetric across the x-axis b. Symmetric across the y-axis c. Symmetric about the origin

(Graphing program optional.) A function is said to be even if (x) (x) and odd if (x) (x) for all x in s domain. Use these definitions to: a. Show that the even integer power functions are even. b. Show that the odd integer power functions are odd. c. Show whether each of the following functions is even, odd, or neither. i. (x) 5 x 4 1 x 2 iii. ii. u(x) 5 x 5 1 x 3 iv. g(x) 5 10.3x d. For each function that you have identified as even or odd, what would you predict about the symmetry of its graph? If possible, check your predictions with a function graphing program. 8.

Use the function (x) to create a new function g(x) where the graph of g(x) is: a. The graph of (x) shifted 3 units to the left, then multiplied by 5, and finally shifted down by 4 units. Does the order of the transformations matter? b. The graph of (x) shifted 3 units to the right, multiplied by 5, then finally shifted up by 4 units. Does the order of the transformations matter?

For each function, construct a new function whose graph is the graph of the original function shifted left by two units, then multiplied by , and then shifted down by five units. a. b. g(x) 5 12x3 c. y 5 log x

If write the equation for g(x) that represents each of the following transformations of f(x). a. f(x 2 3) 1 5 b. c. f (x2 1 2) d. f(x 1 a) 2 f(a) where a is a constant

Write an equation for each function, g, h, and j, based on f(x) x. a. The graph of g is the graph of f(x) shifted 12 units to the left. b. The graph of h is the graph of f(x) shifted 3.8 units to the right. c. The graph of j is the graph of f(x) shifted 9 units left and 12 units up.

Write an equation for each function, g, h, and j, based on f (x) x2 . a. The graph of g is the graph of f(x) shifted 8 units left. b. The graph of h is the graph of f(x) shifted 10 units right. c. The graph of j is the graph of f(x) shifted 4 units left and 12 units down.

The equation for Graph A is . Assuming all of the graphs are the same shape, what th

If h(x) f(x 5), a. What changes are made to the input and output of f(x) to create a new function h(x)? b. Does the order of the transformations matter in part (a)? c. Describe the graph of h(x) in relation of f(x).

Apply the transformations specified in parts (a)(e) to f(x) ln x. f(x 1 2) 2 f(x) 2f(x) f(2x) 2f(2x)

(Graphing program optional.) a. Starting with the function f(x) ex , create a new function g(x) by performing the following transformations. At each step show the transformation in terms of f(x) and ex , i. First shift the graph of to the left by 3 units, ii. Then vertically compress your result by a factor of 1/4, iii. Next reflect it across the x-axis, iv. And finally shift it up by 5 units to create b. Graph f(x) and g(x) on the same grid.

Given function f(x) 10 5x , find the function g(x) if: a. The graph of g is the graph of f reflected across the x-axis. b. The graph of g is the graph of f reflected across the y-axis. c. The graph of g is the graph of f reflected across both the x-axis and the y-axis.

The solid-line graphs below represent exponential functions in the form f(x) Cax . Find an equation for each of the solidline graphs. Then use the reflection properties to write an equation for each of the graphs with the dotted lines.

a. Given describe the transformations that created Find b. Use your knowledge of properties of logarithms to find any vertical and horizontal intercepts for the function g(x).

The following two graphs show the hours of daylight during the year for two different locations. One is for a latitude of 40 degrees above the equator (in the Northern hemisphere), and the other for a latitude of 40 degrees below the equator (in the Southern hemisphere). a. Which graph is associated with which hemisphere? b. Using the language of function transformation, describe Graph B in terms of Graph A.

(Graphing program required.) If an object is put in an environment with a fixed temperature A (the ambient temperature), then the objects temperature, T, at time t is modeled by Newtons Law of Cooling: T 5 A 1 Ce2kt, where k is a positive constant. (Note that T is a function of t and as then , so the temperature T of the object gets closer and closer to the ambient temperature, A.) A corpse is discovered in a motel room at midnight. The corpses temperature is 808 and the room temperature is 608. Two hours later the temperature of the corpse had dropped to 758. (Problem adapted from one in the public domain site S.O.S. Math.) a. Using Newtons Law of Cooling, construct an equation to model the temperature T of the corpse over time, t, in hours since the corpse was found. b. Then determine the time of death. (Assume the normal body temperature is 98.6.) c. Graph the function from t 5 to t 5, and identify when the person was alive, and the coordinates where the temperature of the corpse was 98.6, 80, and 75.

(Graphing program required.) Newtons Law of Cooling (see Exercise 21) also works for objects being heated. Suppose you place a frozen pizza (at 32) into a preheated oven set at 350. Thirty minutes later the pizza is at 320ready to eat. a. Determine the constants A, C, and k in Newtons Law. b. Sketch a graph of your function. c. From your graph, estimate when the pizza will be at 200 and then calculate the time. d. According to your model, if you kept the pizza in the oven indefinitely, would the pizza ever reach 350? What would be a reasonable domain for the function as a model for cooking pizza?

In the accompanying figure, match the graphs labeled A, B, and C with graphs of the form P(x), 2 P(x), and 0.5 P(x).

The accompanying graph shows the height of sunflowers over time. Assume H(t) represents the sunflower height (in cm) as a function of t days under normal conditions. a. If we add fertilizer to the plants, the plants grow faster by a factor of C. Mathematically we can describe the height of the fertilized plants F(t) over time t as CH(t). Estimate the constant C from the graphs and describe its meaning in context. b. If we forget to water the plants regularly, the plants will not reach their normal height; they will be smaller than normal by a factor of D. So we can describe the height of the underwatered plants U(t) as DH(t). Estimate the constant D and describe its meaning in context.

Given (t) 5 3t 2 1 4t 2 5 and g(t) 5 6t 1 1, find: a. (t) 1 g(t) b. g(t) 2 (t) c. (t) ? g(t) d. (t) g(t)

Given and , find and simplify: a. (m) 1 g(m) c. ( g )(m) e. b. ( 2 g)(m) d. g(m) (m) a f g ? b (m) g(m) 5 23m 2m 2 5 (m) 5 3 m 2 4

Let (x) 5 3x5 1 x and g(x) 5 x2 2 1. a. Construct the following functions. j(x) 5 (x) 1 g(x), k(x) 5 (x) 2 g(x), l(x) 5 (x) g(x) b. Evaluate j(2), k(3), and l(21).

If h(x) (x) g(x) x2 3x 4, what are possible equations for (x) and g(x)? 5.

You own a theater company and you have an upcoming event. a. You decide to charge $25 per ticket. Construct a basic ticket revenue function R(n) (in dollars), where n is the number of tickets sold. b. You need to pay $500 to keep the box office open for ticket sales. Modify R(n) to reflect this. c. You decide to give 30 free tickets to the patrons of your company. Modify your function in part (b) to reflect this.

Many colleges around the country are finding they need to buy more computers every year, not only to replace broken or outmoded computers, but also because of the increasing use of computers in classrooms, labs, and studios. A college administrator is preparing a 5-year budget plan. She anticipates that her college, which now has 120 computers, will have to increase that amount by 40 per year for the next 5 years. She currently pays $1000 per computer, but she expects the costs will go up by 3% per year because of inflation. a. Construct a function N(t) for the number of computers each year as a function of time t (in years since the present). b. Construct a function C(t) for the individual cost of a computer purchased in year t. c. Construct a function that will describe the total cost of the computers each year. d. For year 5, how much money should the budget allow for computers?

A worker gets $20/hour for a normal work week of 40 hours and time-and-a-half for overtime. Assuming he works at least 40 hours a week, construct a function describing his weekly paycheck as a function of the number of hours worked.

Using the accompanying table, evaluate the following expressions in parts (a)(d). (f 1 g)(2) (f ? g)(3) f (g 2 f)(0) b (1)

Use the table in Exercise 8 to create a new table for the functions a. c. b. j(x) 5 (g 2 f)(x) h(x) 5 (f 1 g)(x) k(x) 5 (f ? g)(x)

(Graphing program optional.) One method of graphing functions is called addition of ordinates. For example, to graph using this method, we would first graph y1 x. On the same coordinate plane, we would then graph . Then we would estimate the y-coordinates (called ordinates) for several selected x-coordinates by adding geometrically on the graph itself the values of y1 and y2 rather than by substituting numerically. This technique is often used in graphing the sum or difference of two different types of functions by hand, without the use of a calculator. a. Use this technique to sketch the graph of the sum of the two functions graphed in the accompanying figure. b. Use this technique to sketch the graph of y x2 x3 for 2 x 2. Then use a graphing tool (if available) and compare. c. Use this technique to graph y 2x x2 for 2 x 5. 1

Using the accompanying graph of f(x) and g(x), find estimates for the missing values in the following table.

From the graph and your results in Exercise 11, find the equations for: a. c. e. b. d. f. a g f g(x) (f 2 g)(x) b (x) f(x) (f 1 g)(x) (f ? g)(x)

The Richland Banquet Hall charges $500 to rent its facility and $40 per person for dinner. The Hall rental requires a minimum of 25 people and a maximum of 100. A sorority decides to hold its formal there, splitting all the costs among the attendees. Let n be the number of people attending the formal. a. Create a function C(n) for the total cost of renting the hall and serving dinner. b. Create a function P(n) for the cost per person for the event. c. What is P(25)? P(100)? What do these numbers represent?

Given the following graphs of f(x) and g(x): a. Draw the graph of ( f g)(x) and ( f g)(x). b. Would the graph of (g f )(x) be the same as the graph of ( f g)(x)? Explain.

The following are graphs of h(x) and j(x). Without using technology, match each combination below with its graph and state the reason for your choice. a. (h j)(x) b. (h j)(x) c. (x)

The accompanying graph gives the production P( y), consumption C(y), and exports (or imports) E(y) of petroleum (in thousands of barrels/day) in Canada for years, y, between 1980 and 2008. a. In both 1980 and 1981, consumption, C(y), exceeded production, P(y). What does that imply about exports (or imports), E(y), during those years? How does the graph verify your answer? b. Estimate the value each of the following: i. P(1980) C(1980) E(1980) ii. P(2000) C(2000) E(2000) iii. P(2008) C(2008) E(2008) c. What does P(y) C(y) E(y) represent in terms of petroleum in Canada? d. What would it mean if P(y) C(y) 0? e. What would it mean if P(y) C(y) 0? 17.

Almost all organizations find it necessary to maintain inventories of goods that are costly to maintain. For example, if the dollars were not invested in inventory, they could be used profitably elsewherefor salaries, new production, and so on. These dollars are called holding costs. On the other hand, carrying a small inventory means ordering more often, which also incurs costs, called ordering (or made-to-order) costs. The graph below shows the behavior of the holding costs H(q), ordering costs O(q), and total inventory costs T(q) as functions of q, the quantity of the order. a. Use the graph to create a function for T(q) in terms of H(q) and O(q). b. The economic order quantity, q*, is the intersection of H(q) and O(q). What happens at this intersection point? What do you think this means in terms of the total cost?

In Chapter 7 we learned about even and odd power functions and in Section 9.1, Exercise 7, we learned that A function f(x) is even if f(x) f(x). A function f(x) is odd if f(x)  f(x). a. Using what you know about reflections and symmetry, describe how the graphs of even and odd functions differ. b. Are the following functions even, odd, or neither? i. N(x) x3 iii. M(x) 5x ii. F(x) x2 iv. G(x) |x| 1

Use the definition of even and odd functions in Exercise 18 to verify the following statements. a. The sum of two even functions is an even function. b. The sum of two odd functions is an odd function. c. The product of two even functions is an even function. d. The product of two odd functions is an even function. e. The product of an even function and an odd function is an odd function.

The accompanying graph gives the annual sales S(t) and profit (or loss) P(t) for Apple for year t from 2002 to 2009. a. What would S(t) P(t) represent? Describe the part of the graph that represents S(t) P(t). b. What would [P(t)/S(t)] 100% represent? S

In Section 1.2, p. 17, Exercise 1, there is a graph about AIDS diagnoses and deaths from 1981 to 2007. If t year, A(t) number of people diagnosed with AIDS, and D(t)  number of people who died from AIDS, both in year t, what would A(t) D(t) represent? How could you depict this on the graph?

When considering a career path in a particular job sector, one might examine the growth (or decline) of that sector of the job market. The following graphs illustrate the growth in Education & Health Services and the Leisure & Hospitality sector, compiled by the Federal Reserve. Both sectors appear to be growing exponentially. Assuming t years since 1939, regression analysis gives: For all employees in Leisure & Hospitality: For all employees in Education & Health: EEHS(t) 5 1348.6e0.039t thousand jobs (with cc 5 0.998) ELAH(t) 5 1932.5e0.029t thousand jobs (with cc 5 0.997) a. Construct formulas for each of the combinations below and then describe what each function would represent in the context of job sectors. b. Over what interval(s) would the graphs of the new functions be increasing or decreasing? Positive or negative? c. Which (if any) of the combinations can be simplified into a single exponential equation? d. If the trend continues, will there be a time when there will be two Education & Health sector jobs for every one Leisure & Hospitality sector job? i. Use one of your formulas to estimate if and when this will occur. ii. Did this prediction use interpolation or extrapolation? How will this affect the accuracy of the prediction?

. Identify which of the following are polynomial functions and, for those that are, specify the degree. a. y 5 3x 1 2 d. y 5 3x 2 2 b. y 5 2 2 x3 e. y 5 3x5 2 4x3 2 6x2 2 12 c. f. y 5 6(x)(x 2 5)(2x 1 7)

Identify the degree of any of the following functions that are polynomials, and for those that are not polynomials, explain why. a. c. y 5 x5/3 1 x2 y 5 3x2 2 2 y 5 2t3 1 5t y 5 2x 4 1 "2 5 1 x4

Evaluate the following expressions for x 5 2 and x 5 22. a. x23 b. 4x23 c. 24x23 d. 24x

Evaluate the following polynomials for x 5 2 and x 5 22, and specify the degree of each polynomial. a. y 5 3x2 2 4x 1 10 c. y 5 22x4 2 x2 1 3 b. y 5 x3 2 5x2 1 x 2 6

Match each of the following functions with its graph at the top of the next column. a. y 5 2x 2 3 b. y 5 3(2x ) c. y 5 (x2 1 1)(x2 2 4)

Match each of the following functions with its graph. a. f(x) 5 x2 1 3x 1 1 c. f(x) 5 2 x3 1 x 2 3 b. f(x) 5 x3 1 x 2 3

For each of the graphs of polynomial functions, at the top of the next page, determine (assuming the arms extend indefinitely in the indicated direction): i. The number of turning points ii. The number of x-intercepts iii. The sign of the leading term iv. The minimum degree of the polynomial v. Estimated maximum value (if any) vi. Estimated minimum value (if any)

Describe how g(x) and h(x) relate to (x). (x) 5 x5 2 3x2 1 4 g(x) 5 2x5 1 3x2 2 4 h(x) 5 x5 2 3x2 2 2

Divide the following functions into groups having the same global shape for large values of x. Explain your groupings. a. y 5 (x2 2 3)(x 1 9) b. y 5 2x4 1 3 c. y 5 x5 2 3x4 2 11x3 1 3x2 1 10x d. y 5 x(3 2 x)(x 1 1)2 e. y 5 7x3 2 3x2 2 20x 1 5 f. y 5 2(x2 1 1)(x2 2 4)

Estimate the maximum number of turning points for each of the polynomial functions. If available, use technology to graph the function to verify the actual number. a. y 5 x 4 2 2x 2 2 5 c. y 5 x 3 2 3x 2 1 4 b. y 5 4t 6 1 t 2 d. y 5 5 1 x

(Graphing program optional.) Describe the behavior of each polynomial function for large values (positive or negative) of the independent variable and estimate the maximum number of turning points. If available, use technology to verify the actual number. a. y 5 22x4 1 4x 1 3 b. y 5 (t 2 1 1)(t 2 2 1) c. y 5 x 3 1 x 1 1 d. y 5 x 5 2 3x 4 2 11x 3 1 3x 2 1 10x

(Graphing program required.) Find the vertical intercept and estimate the maximum number of horizontal intercepts for each of the polynomial functions. Then graph the function using technology to find the actual number of horizontal intercepts (See Exercise 10.) a. y 5 x4 2 2x2 2 5 c. y 5 x3 2 3x2 1 4 b. y 5 4t 6 1 t 2 d. y 5 5 1 x

(Graphing program required.) Find the vertical intercept and estimate the maximum number of horizontal intercepts for each of the polynomial functions. Then, using technology, graph the functions to find the approximate values of the horizontal intercepts. (See Exercise 11.) a. y 5 22x2 1 4x 1 3 b. y 5 (t 2 1 1)(t 2 2 1) c. y 5 x3 1 x 1 1 d. y 5 x5 2 3x4 2 11x3 1 3x2 1 10x

a. (Graphing program required.) Use a function graphing program to estimate the x-intercepts for each of the following. Make a table showing the degree of the polynomial and the number of x-intercepts. What can you conclude? y 5 2x 1 1 y 5 x3 25x2 1 3x 1 5 y 5 x2 2 3x 2 4 y 5 0.5x4 1 x3 2 6x2 1 x 1 3 b. Repeat part (a) for the following functions. How do your results compare with those for part (a)? Are there any modifications you need to make to your conclusions in part (a)? y 5 3x 1 5 y 5 x3 22x2 2 4x 1 8 y 5 x 2 1 2x 1 3 y 5 (x 2 2)2(x 1 1)2

(Graphing program required.) Use a function graphing program (and its zoom feature) to estimate the number of x-intercepts and their approximate values for: a. y 5 3x3 2 2x2 2 3 b. f(x) 5 x2 1 5x 1 3

(Graphing program required.) Identify the x-intercepts of the following functions, then graph the functions to check your work. a. y 5 3x 1 6 c. y 5 (x 1 5)(x 2 3)(2x 1 5) b. y 5 (x 1 4)(x 2 1)

a. If the degree of a polynomial is odd, then at least one of its zeros must be real. Explain why this is true. b. Sketch a polynomial function that has no real zeros and whose degree is: i. 2 ii. 4 c. Sketch a polynomial function of degree 3 that has exactly: i. One real zero ii. Three real zeros d. Sketch a polynomial function of degree 4 that has exactly two real zeros.

In each part, construct a polynomial function with the indicated characteristics. a. Crosses the x-axis at least three times b. Crosses the x-axis at 1, 3, and 10 c. Has a y-intercept of 4 and degree of 3 d. Has a y-intercept of 4 and degree of 5

Polynomial expressions of the form an3 bn2 cn d can be used to express positive integers, such as 4573, using different powers of 10: 4573 5 4?1000 1 5?100 1 7?10 1 3 or 4573 5 4?103 1 5?102 1 7?101 1 3?100 or if n 5 10, 4573 5 4?n3 1 5?n2 1 7?n1 1 3?n0 5 4n3 1 5n2 1 7n 1 3 Notice that in order to represent any positive number, the coefficient multiplying each power of 10 must be an integer between 0 and 9. a. Express 8701 as a polynomial in n assuming n 10. b. Express 239 as a polynomial in n assuming n 10. Computers use a similar polynomial system, using binary numbers, to represent numbers as sums of powers of 2. The number 2 is used because each minuscule switch in a computer can have one of two states, on or off; the symbol 0 signifies off, and the symbol 1 signifies on. Each binary number is built up from a row of switch states, each set at 0 or 1 as multipliers for different powers of 2. For instance, in the binary number system 13 is represented as 1 1 0 1, which stands for 1?23 1 1?22 1 0?21 1 1?20 5 1?8 1 1?4 1 0?2 1 1?1 5 8 1 4 1 1 5 13 c. What number does the binary notation 1 1 0 0 1 represent? d. Find a way to write 35 as the sum of powers of 2; then give the binary notation.

(Graphing program required.) A manufacturer sells childrens wooden blocks packed tightly in a cubic tin box with a hinged lid. The blocks cost 3 cents a cubic inch to make. The box and lid material cost 1 cent per square inch. (Assume the sides of the box are so thin that their thickness can be ignored.) It costs 2 cents per linear inch to assemble the box seams. The hinges and clasp on the lid cost $2.50, and the label costs 50 cents. (See the accompanying figure.) a. If the edge length of the box is s inches, develop a formula for estimating the cost C(s) of making a box thats filled with blocks. b. Graph the function C(s) for a domain of 0 to 20. What section of the graph corresponds to what the manufacturer actually producesboxes between 4 and 16 inches in edge length? c. What is the cost of this product if the cubes edge length is 8 inches? d. Using the graph of C(s), estimate the edge length of the cube when the total cost is $100.

Factor the following to determine the horizontal intercepts. Then draw a possible sketch to determine the number of turning points and the global behavior. a. f(x) x3 3x2 c. h(x) 2x3 x2 15x b. g(x) x4 81 22.

Factor the following to determine the horizontal intercepts. Then draw a possible sketch to determine the number of turning points and the global behavior. a. f(x) 12x2 23x 10 b. g(x) 3x5 48x c. g(x) x3 3x2 5x 15 (Hint: Factor by grouping terms.) A cu

Construct a polynomial function with the following x-intercepts and with the indicated end behavior. Draw a possible sketch of its graph. a. x 3, x 2, and x 6; and as and as a. x 3, x 2, and x 6; and as and as x S 2`, y S 2` b. x 5, x 2, x 1, and x 3; and as and a x S 1`, y S 2` x S 2`, y S 2`

Construct three third-degree polynomials, each of which has horizontal intercepts at x 2, x 3, but vary by going through the given points: a. f(x) goes through the point (0, 10). b. g(x) goes through the point (1, 48). c. h(x) goes through the point (1, 4). 25.

Construct a polynomial function for each of the following sets of properties and then sketch its graph. The functions should be in the form f(x) a(x r1)(x r2)(x r3) . . . (x rn). (Hint: Find a.) a. Is of degree 3, has a leading coefficient of 2, and has horizontal intercepts at x 1, x 2, and x 4. b. Is of fourth degree, goes through the point (0, 8), and has horizontal intercepts at x 1 and x 2. c. Is of fifth degree with zeros at x 1 twice, x i, and x 0 and with a leading coefficient of a 1. (Remember that i is the complex number .) d. Has the same characteristics as in part (c), but has the leading coefficient of a 1. 26. Are the only

Are the only functions with x-intercepts at 0, 2, and 1 of the form kf(x), where k is a constant and f(x) x(x 2) (x 1)? 2

Match each polynomial function with its graph. f(x) (x 3)2 (x 2)2 g(x) (x 3) (x 2)2 h(x) (x 3)3 (x 2)4 28. A

An ice bucket is designed as a cubic block of foam with a centered cylindrical ice cavity. The side of the outer cube is s inches. Because there is 1minimum thickness of foam at each side of the cube, the diameter of the cylindrical cavity is s 2. The bottom of the cavity is 1from the bottom of the cube. There is no lid. a. Find the radius of the ice cavity, then develop a formula for the volume, V, of rigid foam required for this ice bucket, as a function of s. Recall that the volume of a cylinder of radius r and height h is r2h. In this context, the height is s 1. b. Make a graph of V for s, values from 5to 15. How much rigid foam is required for the 10cube design? 29.

Professional fund-raisers typically establish fund-raising categories for a range of donation amounts. They expect few very large donations, but increasing numbers of donors as the donation categories decrease in amount. A school plans the following fund-raising levels: Lead donor: $25,000; Sponsor: $5000; Sustainer: $1000; Supporter: $200; Friend: $40. a. Find a formula of the form y Cbn, where b 5 and n varies from 0 to 4, that generates each of these five gift amounts. (Hint: Donor amounts are in multiples of 5.) b. The school aims to get 2 lead donors, 10 sponsors, 40 sustainers, 200 supporters, and 500 friends. How much money is it planning to raise? c. Using the formula in part (a), for each term, write an equation for the total money M to be donated in the form: where C is a constant. What value of b fits the school donation plan? d. Plot your formula for values of b from 1 to 10. How much money is raised if b 10?

(Graphing program required.) Buildings lose heat in three different ways: transmission through the building exterior walls and roof, HT; cold air infiltration through cracks and openings, HI; and heat lost from ventilators, HV. Here we look at heat loss for a cubic building with length, width, and height each equal to L feet. The area of each exterior surface of the cube  L L L2 , and the volume of the cube L L L L3. The total heat loss is The transmission loss is directly proportional to the exposed area of the building exterior, which is 5L2, where 5L2 is the 4 walls L2 for the roof. Given a transmission loss constant, a, we have: The infiltration loss comes from having to heat up the volume of cold air that leaks in when the warm air leaks out. It is H proportional to the full interior volume of the house, which is L3 , so given an infiltration loss constant, b, we have: The ventilation loss depends on the strength of any ventilators or hoods that exhaust room air. It is not dependent on volume or area of the house, so given a ventilation loss constant, c, we have: So the total loss becomes For a typical older, moderately drafty house in the northeastern United States with no mechanical ventilation, and c 0. So for this case we have Htotal 0.330(5L2 ) 0.0216(L3 ) This gives heat loss as the temperature difference from inside to outside, measured in degrees Fahrenheit per hour. a. Graph HT, HI, and Htotal on the same graph for L values of 10to 80on the horizontal axis. b. Infiltration losses are typically anywhere up to a third or a half of the heat loss. What is the percent infiltration relative to transmission loss for L 20and L 50? c. Now look at two more buildings with the same volume, L3 , but different exposed exterior areas. Calculate exposed area (walls and roof only) and create Htotal formulas for both. Make a graph showing all three buildings with the same volume, for L from 10to 80. How does changing the area enclosing the building volume affect the heat loss? L L L L L /2 2L L L /2 2L Cubic Building Long Building Tall Building Tall building: L ? L/2 with height 2L Long building: 2L ? L/2 with height L Cubic building: L ? L with height L

If the cost for a producer to produce n items is C(n) 1000 100n, then: a. Create a function F(n) for the cost per item, b. What is C(10)? C(100)? C(1000)? c. What is F(10)? F(100)? F(1000)? d. What trend do these values indicate?

The start-up costs for a small pizza company are $100,000 (the fixed cost) and it costs $3 to produce each additional pizza (the marginal cost). a. Construct a function C(x) for the total cost of producing x pizzas. b. Then create a function P(x) for the total cost per pizza. c. As more pizzas are produced, what happens to the cost per pizza?

Use a calculator to evaluate each expression at x 10, x 100, and x 1000. Then use your answers to estimate the equation of the horizontal asymptote of the function. Specify the domain of each function. a. b. G(x) 5 2x2 2 8 x2 1 x 2 6 F(x) 5 3x 2 2 3x 1 5 4

Use a calculator to evaluate each function at x 10, x 100, and x 1000. Then explain why there is or is not a horizontal asymptote. Estimate the equation of the horizontal asymptote (if it exists) of the function. Find the domain and range. a f(x) 5 x 2 3 (x 2 1)2(x 1 2) b. g(x) 5 x2 1 2x 2 8 4x2 2 9 c. h(x) 5 x2 1 2x 2 3 2x 2 1

Determine the coordinates of the point(s) where the graph of the function y1 intersects its horizontal asymptote y2. Find the domain and range. a. y1 5 3x2 1 x 1 2 x2 1 1 y2 5 3 b. y1 5 2x2 2 8 2x2 1 x 2 6 y2 5 1 c. y1 5 x2 1 2x 2 2 4x2 2 1 y2 5 1 4

If , find f(x) 2f(x 3); then find a common denominator and combine into one rational expression. Specify the domain and range.

For each rational function graphed below, estimate the equation for any vertical or horizontal asymptote(s).

(Graphing program optional.) For each of the functions in Exercises 810, identify any horizontal intercepts and vertical asymptotes. Then, if possible, use technology to graph each function and verify your results. f(x) 5 3x 2 13 x 2 4

(Graphing program optional.) For each of the functions in Exercises 810, identify any horizontal intercepts and vertical asymptotes. Then, if possible, use technology to graph each function and verify your results. g(x) 5 2 2 (x 1 3)2

(Graphing program optional.) For each of the functions in Exercises 810, identify any horizontal intercepts and vertical asymptotes. Then, if possible, use technology to graph each function and verify your results. h(x) 5 1 (x 1 3)(x 2 1) 2 2

(Graphing program required.) a. What is the domain of the rational function b. Does the function have any horizontal intercepts? Any vertical asymptotes? c. What is its end behavior? d. Graph the function. The graph is one of a set of curves called serpentine by Isaac Newton. Why would that name be appropriate?

(Graphing program required.) Let a. We can think of g(x) as being created from transformations of the function Describe the transformations and then write g(x) as a function of f(x). b. Show that g(x) is a rational function of the form (Hint: Find a common denominator.) c. Identify any horizontal intercepts and any vertical asymptotes. d. What is its end behavior? e. Use graphing technology to confirm your answers and estimate the horizontal asymptote.

(Graphing program required.) Construct a rational function f(x) that has horizontal intercepts at (3, 0) and (4, 0) and vertical asymptotes at the lines x 1 and x 5. Use technology to sketch the graph of f(x). What is its end behavior? p(

The function f(x) 1/x was transformed into the function g(x) plotted on the accompanying graph. Construct g(x) in terms of f(x) and then write g(x) in rational function form What is its end behavior?

Without using technology, match each function with its graph. a. f(x) 5 2 2 3(x 1 2) 1 2 b. g(x) 5 3x 1 5 x2 2 4 c. h(x) 5 x2 2 9 5x 2 20

(Graphing program required for part (c).) The rational function can be decomposed into a sum by using the following method: write as sum of a fraction and a constant find the common denominator multiply and simplify Set numerators equal 4x 2 11 5 Bx 1 A 2 3B Set x values equal 4x 5 Bx then B 5 4 Set the constants equal 211 5 A 2 3B substitute 4 for B 211 5 A 2 3(4) then A 5 1 Therefore g(x) 5 4x 2 11 x 2 3 5 1 x 2 3 1 4 4x 2 11 x 2 3 5 A 1 Bx 2 3B x 2 3 4x 2 11 x 2 3 5 A x 2 3 1 B(x 2 3) x 2 3 4x 2 11 x 2 3 5 A x 2 3 1 B which is the graph of f(x) 5 shifted to the right by three units, then shifted up by four units. See the accompanying graph. a. Use the preceding method to decompose . b. Describe the steps in transforming the function into . c. Using technology, plot the graphs of f(x) and g(x) to verify that the transformation described in part (b) is correct.

If , a. Describe the transformations of f(x) used to create the new functions g(x), h(x), and k(x). b. Determine the domain of each function in part (a). c. Determine the equation of the vertical asymptote for each function in part (a). d. Describe the end behavior of each function.

Let . Construct a new function j(s) that is the end result of the transformations of the graph of k(s) described in the following steps. Show your work for each transformation. a. First shift k(s) to the right by two units. b. Then compress your result by a factor of 1/3. c. Reflect across the s-axis. d. Finally, shift it up four units.

(Graphing program required.) You live in a house that borders a river. You want to construct a large garden of 400 square feet along the river which is surrounded by a fence on three sides (excluding the riverfront) to try to keep out the rabbits and deer. Find the dimensions that require the least amount of fence using the following strategy. a. If x and y equal the dimensions (in feet) of the garden, describe the area of the garden. Then solve for y in terms of x. b. Describe the length of fence, F, needed in terms of x and y. c. Substitute the expression for y in part (a) into the equation for part (b) to get a function for F in terms of x. Then rewrite the function as the quotient of two polynomials. (Hint: Find the common denominator.) d. Use technology to Graph F and then estimate the minimum value for F giving the shortest value needed for the fence.

A student drives non-stop from Missoula, Montana to Spokane, Washington. The trip consists of 110 miles of travel in Montana and 90 miles of travel in Idaho and Washington. The speed limit is 75 mph in Montana and is 65 in Idaho and Washington. a. If the student drives at the speed limit (with no stops), how long would the trip take? b. Construct an expression for T(x) which represents the driving time if the students drives x mph over the speed limit. Rewrite T(x) as the quotient of two polynomials. c. Find T(0) and compare with your answer in part (a). d. Find T(10). Is the driving time shorter or longer than T(0)? Why? e. Find T(5). Is the driving time shorter or longer than T(0)? Why? Note: You can get a ticket if you travel at a speed that is too much above or below the posted speed limit.

The graduate math department at an East Coast university subsidizes graduate students who attend the dinners held after each guest lecture. The individual dinner cost is computed by dividing the bill, including tip, by the number of diners minus 1, because the guest lecturer does not pay. The faculty members each pay the full individual dinner cost, and the graduate students each pay half the individual dinner cost. The math department pays the rest, including the cost of the guest and half the cost for each graduate student. There are 3 faculty members who always attend the lectures and g graduate students, which ranges from 10 to 40. So the total number of diners including the lecturer is 4 g, but the number of payers is only 3 g. Assume the restaurant charges $20 for dinner then, The bill per person is $20 20% tip $20 $4 $24. The total bill is $24 times the total number of diners: B 24(g 4). The bill per payer, P, is the total bill, B, divided by the number of payers, g 3, is Faculty members each pay P dollars, the full individual dinner cost. Graduate students each pay half the individual dinner cost: . The total amount A, paid by all the dinersthe 3 faculty and g grad studentsis A 5 3 ? 24 g 1 4 g 1 3 1 g ? 12 g 1 4 g 1 3 Pgrad 5 12 g 1 4 g 1 3 P 5 B g 1 3 5 24 (g 1 4) g a. Simplify the expression for A. b. The amount M paid by the math department is the total cost of the dinner minus the total amount A paid by faculty and students. Simplify the expression for M. c. We could also think of M as the amount that the math department is subsidizing the dinners for the graduate students. Verify that. d. If there are 10 grad students attending the lecture, do they each pay more or less than if 40 attended? e. How is the M formula different if the guest lecturer brings a spouse or other guest who also does not pay?

Building with energy-saving walls becomes increasingly important as fossil fuels become more expensive and less available. To greatly reduce heat loss and energy consumption, passive solar houses are now built with very thick insulated walls, up to 18 inches thick. The amount of heat transmitted through a wall assembly is where U is the heat transmitted in British thermal units, Btu, per square foot, per hour, per degree of Fahrenheit temperature difference from inside to outside. RTotal is the sum of the resistances to heat transfer through each layer of the wall materials from inside to outside. The total heat transmitted is the reciprocal of the total resistance to heat transmission. When resistance is low, the transmission is high, and vice versa. A typical wall assembly, asshown in the crosssection below, has exterior and interior surface materials mounted on wood studs spaced 16 inches apart inside the wall. The space between the studs is filled with insulation, usually fiberglass or rigid foam. Because more heat is lost through the studs than through the insulated parts of the wall, a stud is called a thermal bridge. Outside Wall Cross Section Plywood sheathing Studs Rigid foam insulation Gypsum wall board Clapboards siding U 5 1 RTotal M 5 g ? 12 g 1 4 g 1 3 M 5 24(g 1 4) 2 A. 9.5 Composition and Inverse Functions 585 The resistance total for the inside and outside layers (of clapboards, plywood and gypsum) is the same all along the wall, about 3. The resistance of a stud alone is 1.25d for d inches of depth. The resistance of rigid foam insulation is 5d. Since studs are doubled beside windows and doors, approximately 15% of a wall has studs and the remaining 85% is insulated, so we have a. Rearrange and simplify the UWall formula so the terms have a common denominator. b. Graph your formula using d on the horizontal axis from 0 inches to 18 inches (0!to 18!). c. Studs are always smaller than their names suggest because the finished lumber has been planed. A typical wood house with nominal 2!by 4!studs has d "3.5!. A new wood house with nominal 2!by 6!studs has d "5.5!. A passive solar house with nominal 2!by 12!studs has d "11.5!. i. What are the U values for heat transmission in these three houses? ii. What is the percent of energy saved relative to the typical house for the new house and for the passive solar house? d. A Solar Decathlon house, designed for the biennial Department of Energy competition in Washington, D.C., is built with 24!stud spacing and d "15!. The studs now occupy only about 10% of the wall. Develop the UWall heat transmission value for this house. i. What percent energy saving does the Decathlon house wall give relative to the typical house? ii. Does making d "18!instead of 15!give a big improvement?

From the accompanying table, find: a. f(g(1)) c. f(g(0)) e. f( f(2)) b. g( f(1)) d. g( f(0))

From the accompanying table, find: a. f(g(1)) c. f(g(0)) e. f( f(2)) b. g( f(1)) d. g(f(0))

Using the accompanying graphs, find: a. g(f(2)) b. f(g(21)) c. g(f(0)) d. g(f(1))

Using the accompanying graphs, find: a. g(f(22)) b. f(g(1)) c. g(f(0)) d. g(f(1))

Given F(x) 5 2x 1 1 and , find: a. F(G(1)) d. F(F(0)) b. G(F(22)) e. (F G)(x) c. F(G(2)) f. (G F)(x) s s G(x) 5 x 2 1 x 1 2

Given f(x) 5 3x 2 2 and g(x) 5 (x 1 1)2 , find: a. f(g(1)) d. f(f(2)) b. g(f(1)) e. (f g)(x) c. f(g(2)) f. (g f)(x)

The winds are calm, allowing a forest fire to spread in a circular fashion at 5 feet per minute. a. Construct a function A(r) for the circular area burned, where r is the radius. Identify the units for the input and the output of A(r). b. Construct a function for the radius r R(t) for the increase in the fire radius as a function of time t. What are the units now for the input and the output for R(t)? c. Construct a composite function that gives the burnt area as a function of time. What are the units now for the input and the output? d. How much forest area is burned after 10 minutes? One hour?

The exchange rate a bank gave for Canadian dollars on June 27, 2010, was 0.961 Canadian dollars for 1 U.S. dollar. The bank also charges a constant fee of 3 U.S. dollars per transaction. a. Construct a function F that converts U.S. dollars, d, to Canadian dollars. b. Construct a function G that converts Canadian dollars, c, to U.S. dollars. c. What would the function F G do? Would its input be U.S. or Canadian dollars (i.e., d or c)? Construct a formula for F G. d. What would the function G F do? Would its input be U.S. or Canadian dollars (i.e., d or c)? Construct a formula for G F.

A stone is dropped into a pond, causing a circular ripple that is expanding at a rate of 13 ft sec. Describe the area of the circle as a function of time.

The wind chill temperature is the apparent temperature caused by the extra cooling from the wind. A rule of thumb for estimating the wind chill temperature for an actual temperature t that is above 0 Fahrenheit is W(t) t 1.5S0, where S0 is any given wind speed in miles per hour. a. If the wind speed is 25 mph and the actual temperature is 10 F, what is the wind chill temperature? We know how to convert Celsius to Fahrenheit; that is, we can write t F(x), where , with x the number of degrees Celsius and F(x) the equivalent in degrees Fahrenheit. b. Construct a function that will give the wind chill temperature as a function of degrees Celsius. c. If the wind speed is 40 mph and the actual temperature is 10 C, what is the wind chill temperature? 11

Salt is applied to roads to decrease the temperature at which icing occurs. Assume that with no salt, icing occurs at 32 F, and that each unit increase in the density of salt applied decreases the icing temperature by 5 F. a. Construct a formula for icing temperature, T(s), as a function of salt density, s. Trucks spread salt on the road, but they do not necessarily spread it uniformly across the road surface. If the edges of the road get half as much salt as the middle, we can describe salt density s S(x) as a function of the distance, x, from the center of the road by where k is the distance from the centerline to the road edges and Sd is the salt density applied in the middle of the road. b. What will the expression for S(x) be if the road is 40 feet wide? c. What will the value for x be at the middle of the 40-footwide road? At the edge of the road? Verify that at the middle of the road the value of the salt density S(x) is Sd and that at the edge the value of S(x) is . d. Construct a function that describes the icing temperature, T, as a function of x, the distance from the center of the 40-foot-wide road. (Hint: Compose (T S)(x).) e. What is the icing temperature at the middle of the 40-footwide road? At the edge?

Using the given functions f, g, and h where f(x) x 1 g(x) 5 ex h(x) x 2 a. Create the function k(x) (f g h)(x). b. Describe the transformation from x to k(x). 13

Using the given functions J, K, and L, where J(x) x3 K(x) log(x) L(x) a. Create the function M(x) 5 (L J K)(x). b. Describe how to get from x to M(x). I

In Exercises 14 and 15, rewrite j(x) as the composition of three functions, f, g, and h. j(x) 5 2 (x 2 1)3

In Exercises 14 and 15, rewrite j(x) as the composition of three functions, f, g, and h. j(x) 4ex1

In Exercises 1622, show that the two functions are inverses of each other (x) 5 3x 1 2 and g(x) 5 x 2 2 3

In Exercises 1622, show that the two functions are inverses of each other (x) 5 (where x . 1) and g(x) 5 x2 1 1 (where x . 0)

In Exercises 1622, show that the two functions are inverses of each other (x) 5 2x 2 1 and g(x) 5 x 1 1 2

In Exercises 1622, show that the two functions are inverses of each other f(x) 5 and g(x) 5 x3 2 5 4

In Exercises 1622, show that the two functions are inverses of each other f(x) 5 10x/2 and g(x) 5 log(x2)

In Exercises 1622, show that the two functions are inverses of each other F(t) 5 e 3t and G(t) 5 ln(t 1/3) where t . 0

In Exercises 1622, show that the two functions are inverses of each other H(r) 5 ln r where r . 0 and J(r) 5 e 2r

In Exercises 23 and 24, create a table of values for the inverse of the function (x).

In Exercises 23 and 24, create a table of values for the inverse of the function (x).

Cryptology (the creation and deciphering of codes) is based on 1-1 functions. After you code a message using a 1-1 function, the decoder needs the inverse function in order to retrieve the original message. The following table matches each letter of the alphabet with its coded numerical form. ABCDE FGH I J KLM 26 25 24 23 22 21 20 19 18 17 16 15 14 NO P Q R S T UVWXY Z 13 12 11 10 9 8 7 6 5 4 3 2 1 a. Does this code represent a 1-1 function? Is there an inverse function? If so, what is its domain? b. Decode the message 14 26 7 19 9 6 15 22 8.

On June 27, 2010, the conversion rate from U.S. dollars to euros was 1.229; that is, on that day you could change $1 for 1.229 euros, the currency of the European Union. a. Was a U.S. dollar worth more or less than 1 euro? b. Using the June 27 exchange rate, construct a function C1(d) that converts d dollars to euros. What is C1(1)? C1(25)? c. Now construct a second function C2(r) that converts r euros back to dollars. What is C2(1)? C2(100)? d. Show that C1 and C2 are inverses of each other. e. Reread the beginning of Exercise 8 (in this section), which describes a conversion process between Canadian and U.S. dollars. In that process the two formulas are not inverses of each other. Why not?

Given the accompanying graph of f(x), answer the following. a. Does f(x) have an inverse? Please explain. b. What is the domain of f(x)? Estimate the range of f(x). c. From the graph, determine f(4), f(0), and f(5). d. Determine f 1 (0), f 1 (2), and f 1 (3). 5

Determine which of the accompanying graphs show functions that are one-to-one.

In Exercises 2931, for each function Q find Q1 , if it exists. For those functions with inverses, find Q(3) and Q1 (3). Q(x) 5 23 x 2 5

In Exercises 2931, for each function Q find Q1 , if it exists. For those functions with inverses, find Q(3) and Q1 (3). Q(x) 5 5e0.03x

In Exercises 2931, for each function Q find Q1 , if it exists. For those functions with inverses, find Q(3) and Q1 (3). Q(x) 5 x 1 3 x

Use the graph of f(x) to estimate the value of each expression. a. f(2) b. f 1 (2) c. f 1 (4) d. (f f 1 )(8) 3

The following tables represent a function f that converts cups to quarts and a function g that converts quarts to gallons (all measurements are for fluids). a. Fill in the missing values in the chart. (Hint: One quart contains 4 cups, and one gallon contains 4 quarts.) b. Now evaluate each of the following and identify the units of the results. i. (g f)(8) iii. (f 1 g1 )(1) ii. g1 (2) iv. ( f 1 g1 )(2) c. Explain the significance of (f 1 g1 )(x) in terms of cups, quarts, and gallons. 34. Le

Let f(x) mx b. a. Does f(x) always have an inverse? Explain. b. If f(x) has an inverse, find f 1 (x). c. Using the formula for f 1 (x), explain in words how, given any linear equation (under certain constraints), you can find the inverse function knowing the slope m and y-intercept b.

If you do an Internet search on formulas for ideal body weight (IBW), one that comes up frequently was created by Dr. B. J. Devine. His formula states IBW for men (in kilograms) 50 (2.3 kg per inch over 5 feet) IBW for women (in kilograms) 45.5 (2.3 kg per inch over 5 feet) a. Write the functions for IBW (in kg) for men and women, Wmen(h) and Wwomen(h), where h is a persons height in inches. Give a reasonable domain for each. b. Evaluate Wmen(70) and Wwomen(66). Describe your results in terms of height and weight. c. Evaluate W1 men(77.6). What does this tell you? d. Given that 1 lb 0.4356 kg, alter the functions to create Wnewmen(h) and Wnewwomen(h) so that the weight is given in pounds rather than kilograms. e. Use your functions in part (d) to find W1 newwomen(125). What does this tell you? [Note: More information can be found in the article by M. P. Pari and F. P. Paloucek, The origin of the ideal body weight equations, Annals of Pharmacology 34 (9), 2000: 10 :106669.] 36.

The formula for the volume of a cone is V 5 r2 h. Assume you are holding a 6-inch-high sugar cone for ice cream. a. Construct a function V(r) for the volume as a function of r. Why dont you need the variable h in this case? Find V(1.5) and explain what have you found (using appropriate units). b. Evaluate V1(25). Describe your results. What are the units attached to the number 25? c. When dealing with abstract functions where f(x) y, we have sometimes used the convention of using x (rather than y) as the input to the inverse function f 1 (x). Explain why it does not make sense to interchange V and r here to find the inverse function. 3

In Chapter 6 we learned that a logarithm can be constructed using any positive number (except 1) as a base: logax y means that ay x. Show that F(x) ax and G(x) logax are inverse functions. The software E10: Inverse Functions y ax and y loga x in Exponential and Log Functions can help you visualize the relationship between the two functions. A 1

The percentage of a building exterior that is glass is very important because the highest heat energy losses occur through glass areas. Roofs and walls can be much better insulated than glass and have a much higher resistance to transmission of heat. To estimate the effect of glass percentage G (in decimal form) on heat loss, we look at average resistance Rt for the total building exterior, using Rg for glass resistance to heat transmission and Rw for roof and wall resistance. Heat transmission loss, Ut , through the exterior can then be found using Rt . Ut is the heat loss for 1 square foot per hour per degree Fahrenheit temperature difference from inside to outside of the building exterior, measured in British thermal units. Rt 5 GRg 1 (1 2 G)Rw Ut 5 1 Rt 9.6 Exploring, Extending & Expanding 599 where (1 G) is the decimal percent of the rest of the building exterior, walls, and roof. a. Simplify the formula for Rt , then use it to find a formula for Ut . b. On the same grid, using G as the horizontal axis, plot your heat loss formulas for the following cases: i. Old house with Rg 1 and Rw 10 ii. Typical modern house with Rg 2 and Rw 20 iii. Energy saving house with Rg 5 and Rw 60 c. The typical modern house can be improved by just changing the windows so that Rg 5, Rw 20. On the same graph, plot this improved house with the typical modern house from part (ii) above. Does improving the windows make a substantial reduction in heat loss? 9.6 Expl

a. Sketch the graphs of the following functions on the same grid: f(x) x3 g(x) (x 2)3 h(x) x3 2 b. Write g(x) and h(x) as transformations of f(x).

Using the three accompanying graphs of polynomial functions in the next column, determine whether the degree of the polynomial is odd or even, identify its minimum possible degree, and estimate any visible horizontal intercepts of the function.

Which of the following statements are true about the graph of the polynomial function f(x) x3 bx2 cx d a. It intersects the vertical axis at one and only one point. b. It intersects the x-axis in at most three points. c. It intersects the x-axis at least once. d. The vertical intercept is positive. e. For large (positive or negative) values of x, the graph looks like y x3 . f. The origin is a point on the graph.

a. Generate two different polynomials, M(z) and N(z), that have horizontal intercepts at z 2, 0, and 3. b. Generate a third polynomial, P(z), with the same horizontal intercepts but a higher degree.

On the same grid, hand-draw rough sketches of the three functions f(x) ex , g(x) 4ex , and h(x) 0.5ex for 0 x 5. Describe the relationships among the graphs. 6.

a. Given the following graph of the function f(x), sketch: i. g(x) f(x) ii. h(x) f(x) iii. j(x) |f(x)| b. Describe each function in relation to f(x). 7.

(Graphing program required.) a. If and , describe the transformation of the graph of f(x) into the graph of g(x). Does the order of the transformations matter? b. Rewrite g(x) as a ratio of two polynomials. c. What is the domain of g(x)? Sketch g(x). d. What are its horizontal and vertical intercepts (if any)? e. Does g(x) have a vertical asymptote? f. What is its end behavior?

Global warming melts glaciers and polar ice, so scientists predict that the sea level will rise, flooding coastal areas. a. Assuming Earth is a sphere with radius r and that roughly three-quarters of Earths surface is ocean, develop a formula to estimate the volume of melt water necessary to raise the sea level 1 foot. See image at top of next column. (Note: The volume of a sphere is .) b. Given the Earths radius is currently about 3959 miles and 1 cubic foot 7.481 gallons, how many gallons of melt water does your estimate predict? (Recall: 1 mile 5280 feet.)

Given the graph of f(x) and g(x) below: Find the following values and complete the table.

If f(x) (1/2)x and g(x) (1/2) x (x 4), a. Find expressions for i. f(x) g(x) ii. f(x)/g(x) b. Determine the values in the table. c. Compare your answers in part (b) with the answers in Exercise 9. Explain what you have found. 11

Retirement fund counselors often recommend a mixed portfolio of investments, including some higher-risk investments, which offer higher interest rates, and some more secure investments, with lower interest rates. A woman wants to put half of her $10,000 savings in a safe 4% fund, and the other half in a riskier 10% fund, both compounded annually. She expects to retire in 30 years but would like to know how much she can expect to get if she retires earlier. a. Create three functions where t is the number of years since the start of the investments and S(t) is the amount of money in the 4% account, R(t) is the amount in the 10% account, and T(t) is the total amount invested in both accounts. b. On one graph show how the 4% fund, the 10% fund, and the combined fund total accumulate over 30 years. c. In the worst-case scenario, if she loses all the money in the 10% fund, how much will she be left with in 30 years? d. You might think that if she is getting 4% on $5000 plus 10% on another $5000, this is the same as getting 14% on $5000. Is it? If not, why not? You can explain your answer using a table and/or a graph.

A typical retirement scheme for state employees is based on three things: age at retirement, highest salary attained, and total years on the job. Annual retirement allowance where the maximum percentage is 80%. The highest salary is typically at retirement. We define: total years worked retirement age starting age retirement age factor 0.001 (retirement age 40) salary at retirement starting salary all annual raises a. For an employee who started at age 30 in 1973 with a salary of $12,000 and who worked steadily, receiving a $2000 raise every year, find a formula to express retirement allowance, R, as a function of employee retirement age, A. b. Graph R versus A. c. Construct a function S that shows 80% of the employees salary at age A and add its graph to the graph of R. From the graph estimate the age at which the employee annual retirement allowance reaches the limit of 80% of the highest salary. d. If the rule changes so that instead of highest salary, you use the average of the three highest years of salary, how would your formula for R as a function of A change? 13. F

For the graphs below, use function notation, to express g(x) and h(x) in terms of f(x).

Motorola has taken steps to diversify its business by introducing multiple smartphones, including the popular Droid X and Droid 2. The following chart (from Motorola) describes Motorolas quarterly total sales and mobile device sales. If T(t) total sales (in billions) in quarter t, and M(t) sales (in billions) of mobile devices in quarter t, then what do the following expressions represent? a.T(t) 2 M(t)? b.M(t) T(t) ? 100% c. T(t) 2 M(t) T(t) ? 100%

Match each of the following graphs with its function, find the equation for any vertical or horizontal intercepts, and specify its domain and range. a. c. b. d. h(x) 5 (2x 1 5) (x 2 1) g(x) 5 (x2 2 1) x2 f(x) 5 x2 (x2 2 1) j(x) 5 (2x 1 5)

Given the functions f(x) x2 , g(x) x 1, and h(x) 3x, evaluate each of the following compositions. a. and b. and c. (f + g + h)(x) and (h + g + f)(x) (h + g)(x) (g + h)(x) (f + g)(x) (g + f)(x) c

Most current refrigerators (such as the 2010 Energy Star) cost about $720 and use about 500 kilowatts per hour (kWh) of electricity. In Rhode Island the average (mean) retail price of electricity for consumers in 2010 was 16 cents per kilowatt hour according to the U.S. Energy Information Administration (EIA). a. What is the cost of electricity to operate a 2010 Energy Star refrigerator in Rhode Island for a year? b. Construct a cost equation C(n) for the total cost of buying and operating the refrigerator over n years in Rhode Island. c. Construct an equation for the total cost per year, T(n) C(n)/n. d. What is T (5)? T (10)? T (20)? What happens to the total cost per year as n increases? Sketch the graph of T(n) and explain what it means for the consumer.

(Graphing program required.) Given the rational function , determine the following for parts (a)(e), and then sketch the graph: a. The x and y intercepts b. The vertical and horizontal asymptotes c. The domain and range of the function d. The interval(s) for x where f(x) 0 and where f(x) 0 e. The interval(s) for x where f(x) is increasing and f(x) is decreasing

Does the function f(x) (x 2)3 1 have an inverse? If not, explain why. If so, what is it?

Which of these functions has an inverse? If there is one, what is it?

When lightning strikes, you seem to see it right away, but the associated thunder often comes a few seconds later. One rule of thumb is that each second of delay represents 1000 feet; that is, if you hear the thunder 3 seconds after the lightning strike, the strike was about 3000 feet away from you. a. Light travels about 186,000 miles per second, so the light created from a lightning strike a few thousand feet away is seen virtually simultaneously with the strike. However, sound travels much more slowly, at about 761 mph at sea level. i. Convert 761 mph into feet/second. ii. Now construct a function that gives the distance D(t) (in feet) that the thunder has traveled from the strike site in t seconds. iii. Does the rule of thumb seem reasonable? b. The sound travels in all directions, creating expanding sound circles that radiate out from the lightning strike. i. Create a function A(r) (in square feet) that gives the area of a sound circle with radius r (in feet). ii. We can think of the radius r of the sound circle as D(t), the distance thunder has traveled (in any direction) in t seconds. Substituting r for D(t), construct a composite function A(D(t)) to describe the circular area at time t within which the thunder can be detected. What is the circular area (in square feet) within which thunder can be heard 4 seconds after the lightning strike? What is the area in square miles? iii. When the time doubles, what happens to the distance the thunder has traveled? What happens to the area within which it can be heard?

The Texas Cancer Center website, www.texascancercenter.com, notes that the 5-year survival rate for stage I breast cancer (when the tumor diameter is 2 cm) is about 85%. The 5-year survival rate for stage II breast cancer (when the tumor diameter is 5 cm and the cancer has not spread to the lymph nodes) is about 65%. a. Using the equations in Section 9.6 Example 1 referring to a tumor that doubles in volume every 100 days, how long would it take such a tumor to grow from 0.5 cm in diameter to 5 cm? (Recall that 0.5 cm in diameter corresponds to an initial volume of 0.06 cubic cm, the minimum tumor size detectable by a mammogram.) b. If the tumor were more aggressive, doubling in volume every 50 days, what would the yearly growth factor be? Use this to construct a new function to reflect the volume growth of this more aggressive tumor over time (again using 0.06 cubic cm as the initial volume). c. Using your function from part (b), construct another function to represent the diameter growth over time. d. How many years would it take the aggressive tumor to grow from 0.5 cm to 5 cm in diameter?

For Problems 1725 give an example of a function or functions with the specified properties. Express your answer using equations. A rational function with a horizontal intercept at (2, 0) and two vertical asymptotes at x 0 and x 3.

For Problems 1725 give an example of a function or functions with the specified properties. Express your answer using equations. A function that compresses by a factor of 1/2 and then shifts the resulting function 5 units to the left.

For Problems 1725 give an example of a function or functions with the specified properties. Express your answer using equations. A rational function f(x) with horizontal asymptote y 2, vertical asymptote x 1, and f(x) 0 for all x.

In 2008 the population of C2(t) was approximately half the population of C1(t).

C1(t) C2(t) is a constant.

A polynomial function of degree 4 will always have three turning points.

A polynomial function of degree 4 will cross the horizontal axis exactly four times.

A polynomial function of odd degree must cross the horizontal axis at least one time.

The leading term determines the global shape of the graph of a polynomial function.

The three polynomial functions in the accompanying figure are all of even degree.

The functions f(x) ln x and g(x) ex are inverses of each other.

f 21(x) 5 1 f(x) for any function f

If and g(x) 2x2 1, then (f + g)(x) 5 1 (2x 2 1 1)

If f(x) f(x), the graph of f is symmetric across the x-axis.

If f(x) f(x), the graph of f is symmetric about the origin. 3

A function that passes the vertical line test has an inverse

Every function is 1-1.

The graph below shows the linear path of a ray of light, P, and the corresponding linear path of its reflection, Q. The line Q P.