Solved: Graphical Reasoning An electronically controlled thermostat in a home is

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

The ________ of the function with is

The domain of is all in the domain of such that _______ is in the domain of

To decompose a composite function, look for an ________ function and an ________ function.

In Exercises 512, find (a) (b) (c) and (d) What is the domain of f/g

In Exercises 512, find (a) (b) (c) and (d) What is the domain of f/g

In Exercises 512, find (a) (b) (c) and (d) What is the domain of f/g

In Exercises 512, find (a) (b) (c) and (d) What is the domain of f/g

In Exercises 512, find (a) (b) (c) and (d) What is the domain of f/g

In Exercises 512, find (a) (b) (c) and (d) What is the domain of f/g

In Exercises 512, find (a) (b) (c) and (d) What is the domain of f/g

In Exercises 512, find (a) (b) (c) and (d) What is the domain of f/g

In Exercises 1324, evaluate the indicated function for

In Exercises 1324, evaluate the indicated function for

In Exercises 1324, evaluate the indicated function for

In Exercises 1324, evaluate the indicated function for

In Exercises 1324, evaluate the indicated function for

In Exercises 1324, evaluate the indicated function for

In Exercises 1324, evaluate the indicated function for

In Exercises 1324, evaluate the indicated function for

In Exercises 1324, evaluate the indicated function for

In Exercises 1324, evaluate the indicated function for

In Exercises 1324, evaluate the indicated function for

In Exercises 1324, evaluate the indicated function for

In Exercises 2528, graph the functions and on the same set of coordinate axes.

In Exercises 2528, graph the functions and on the same set of coordinate axes.

In Exercises 2528, graph the functions and on the same set of coordinate axes.

In Exercises 2528, graph the functions and on the same set of coordinate axes.

In Exercises 29 and 30, 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

In Exercises 29 and 30, 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

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

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

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

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

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

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

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

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

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

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

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

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

In Exercises 4346, use the graphs of and to evaluate the functions. (a) (b)

In Exercises 4346, use the graphs of and to evaluate the functions. (a) (b)

In Exercises 4346, use the graphs of and to evaluate the functions. (a) (b)

In Exercises 4346, use the graphs of and to evaluate the functions. (a) (b)

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

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

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

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

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

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

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

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

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 Find the function that represents the total stopping distance Graph the functions and on the same set of coordinate axes for

Sales From 2000 to 2005, the sales (in thousands of dollars) for one of two restaurants owned by the same parent company can be modeled by where represents 2000. During the same six-year period, the sales (in thousands of dollars) for the second restaurant can be modeled by (a) Write a function that represents the total sales of the two restaurants owned by the same parent company. (b) Use a graphing utility to graph and in the same viewing window.

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 2000. (a) If is the population of the United States in year find the function that represents the percent change in the population of the United States. (b) Interpret the value of

Pets 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 2000. (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 2000. Find and interpret

Military Personnel The total numbers of Army personnel (in thousands) and Navy personnel (in thousands) from 1990 to 2002 can be approximated by the models and where represents the year, with corresponding to 1990. (Source: Department of Defense) (a) Find and interpret Evaluate this function for 8, and 12. (b) Find and interpret Evaluate this function for t _ 4, 8, and 12.

Sales The sales of exercise equipment (in millions of dollars) in the United States from 1997 to 2003 can be approximated by the function and the U.S. population (in millions) from 1997 to 2003 can be approximated by the function where represents the year, with corresponding to 1997. (Source: National Sporting Goods Association, U.S. Census Bureau) (a) Find and interpret (b) Evaluate the function in part (a) for 10, and 12.

Health Care Costs The table shows the total amounts (in billions of dollars) spent on health services and supplies in the United States (including Puerto Rico) for the years 1995 through 2001. The variables and represent out-of-pocket payments, insurance premiums, and other types of payments, respectively. (Source: Centers for Medicare and Medicaid Services) (a) Use the regression feature of a graphing utility to find a linear model for and quadratic models for and Let represent 1995. (b) Find What does this sum represent? (c) Use a graphing utility to graph and in the same viewing window. (d) Use the model from part (b) to estimate the total amounts spent on health services and supplies in the years 2008 and 2010.

Graphical Reasoning An electronically controlled thermostat in a home is programmed to lower 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 prepared as 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 _ r__x

Physics A pebble is dropped into a calm pond, causing ripples in the form of concentric circles (see figure). The radius (in feet) of the outer ripple is where is the time in seconds after the pebble strikes the water. The area of the circle is given by the function Find and interpret

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 time when the bacterial count reaches 1500.

Cost The weekly cost of producing units in a manufacturing process is given by The number of units produced in hours is given by (a) Find and interpret (b) Find the time that must elapse in order for the cost to increase to $15,000.

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 given by and If is greater than $500,000, which of the following represents your bonus? Explain your reasoning. (a) (b)

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. True or False?

In Exercises 69 and 70, determine whether the statement is true or false. Justify your answer. If and then

In Exercises 69 and 70, determine whether the statement is true or false. Justify your answer. If you are given two functions and , you can calculate if and only if the range of is a subset of the domain of

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.

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

In Exercises 7376, find the difference quotient and simplify your answer.

In Exercises 7376, find the difference quotient and simplify your answer.

In Exercises 7376, find the difference quotient and simplify your answer.

In Exercises 7376, find the difference quotient and simplify your answer.

In Exercises 7780, find an equation of the line that passes through the given point and has the indicated slope. Sketch the line

In Exercises 7780, find an equation of the line that passes through the given point and has the indicated slope. Sketch the line

In Exercises 7780, find an equation of the line that passes through the given point and has the indicated slope. Sketch the line

In Exercises 7780, find an equation of the line that passes through the given point and has the indicated slope. Sketch the line