Solution: Sketch a rough graph of the market value of a new car as a function of time

If f sxd x 1 s2 2 x and tsud u 1 s2 2 u , is it true that f t?

If fsxd x 2 2 x x 2 1 and tsxd x is it true that f t?

The graph of a function f is given. (a) State the value of fs1d. (b) Estimate the value of fs21d. (c) For what values of x is fsxd 1? (d) Estimate the value of x such that fsxd 0. (e) State the domain and range of f. (f) On what interval is f increasing?

The graphs of f and t are given. (a) State the values of fs24d and ts3d. (b) For what values of x is fsxd tsxd? (c) Estimate the solution of the equation fsxd 21. (d) On what interval is f decreasing? (e) State the domain and range of f. (f) State the domain and range of t.

Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function.

Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function.

Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function.

Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function.

Shown is a graph of the global average temperature T during the 20th century. (a) What was the global average temperature in 1950? (b) In what year was the average temperature 14.28? (c) When was the temperature smallest? Largest? (d) Estimate the range of T.

Trees grow faster and form wider rings in warm years and grow more slowly and form narrower rings in cooler years. The figure shows ring widths of a Siberian pine from 1500 to 2000. (a) What is the range of the ring width function? (b) What does the graph tend to say about the temperature of the earth? Does the graph reflect the volcanic eruptions of the mid-19th century?

A healthy esophagus has a pH of about 7.0. When acid reflux occurs, stomach acid (which has pH ranging from 1.0 to 3.0) flows backward from the stomach into the esophagus. When the pH of the esophagus is less than 4.0, the episode is called clinical acid reflux and can cause ulcers and damage the lining of the esophagus. The graph shows esophageal pH for a sleeping patient with acid reflux. During what time interval is the patient considered to have an episode of clinical acid reflux?

Tadpole weights The figure shows the average body weights of tadpoles raised in different densities. The function f shows body weights when the density is 10 tadpolesyL. For functions t and h the densities are 80 and 160 tadpolesyL, respectively. What do these graphs tell you about the effect of crowding?

Species richness Tropical regions receive more rainfall and intense sunlight and have longer growing seasons than regions farther from the equator. As a result, they enjoy greater species richness, that is, greater numbers of species. The graph shows how species richness varies with latitude for ants. (a) How many species would you expect to find at 308S? At 208N?

In this section we discussed examples of ordinary, everyday functions: Population is a function of time, postage cost is a function of weight, water temperature is a function of time. Give three other examples of functions from everyday life that are described verbally. What can you say about the domain and range of each of your functions? If possible, sketch a rough graph of each function.

The graph shown gives the weight of a certain person as a function of age. Describe in words how this persons weight varies over time. What do you think happened when this person was 30 years old?

The graph shows the horizontal force exerted by the ground on a person during walking. Positive values are forces in the forward direction and negative values are forces in the backward direction. Give an explanation for the shape of the graph of the force function, including the points where it crosses the axis.

You put some ice cubes in a glass, fill the glass with cold water, and then let the glass sit on a table. Describe how the temperature of the water changes as time passes. Then sketch a rough graph of the temperature of the water as a function of the elapsed time.

Three runners compete in a 100-meter race. The graph depicts the distance run as a function of time for each runner. Describe in words what the graph tells you about this race. Who won the race? Did each runner finish the race?

Shown is a typical graph of the number N of bacteria grown in a batch culture as a function of time t. Describe what you think is happening during each of the four phases.

Sketch a rough graph of the number of hours of daylight as a function of the time of year.

Sketch a rough graph of the outdoor temperature as a function of time during a typical spring day.

You place a frozen pie in an oven and bake it for an hour. Then you take it out and let it cool before eating it. Describe how the temperature of the pie changes as time passes. Then sketch a rough graph of the temperature of the pie as a function of time.

Sketch the graph of the amount of a particular brand of coffee sold by a store as a function of the price of the coffee.

Sketch a rough graph of the market value of a new car as a function of time for a period of 20 years. Assume the car is well maintained

Bird count The table shows the number of house finches, in thousands, observed in the Christmas bird count in California. Year 1980 1985 1990 1995 2000 2005 2010 Count 74 92 88 107 70 61 78

Researchers measured the blood alcohol concentration (BAC) of eight adult male subjects after rapid consumption of 30 mL of ethanol (corresponding to two standard alcoholic drinks). The table shows the data they obtained by averaging the BAC (in mgymL) of the eight men. t (hours) 0.0 0.2 0.5 0.75 1.0 1.25 1.5 BAC 0 0.25 0.41 0.40 0.33 0.29 0.24 t (hours) 1.75 2.0 2.25 2.5 3.0 3.5 4.0 BAC 0.22 0.18 0.15 0.12 0.07 0.03 0.01 (a) Use the readings to sketch the graph of the BAC as a function of t. (b) Use your graph to describe how the concentration of alcohol varies with time. Source: Adapted from P. Wilkinson et al., Pharmacokinetics of Ethanol after Oral Administration in the Fasting State, Journal of Pharmacokinetics and Biopharmaceutics 5 (1977): 20724.

If f sxd 3x 2 2 x 1 2, find fs2d, fs22d, fsad, fs2ad, fsa 1 1d, 2fsad, fs2ad, fsa2 d, [ fsad] 2 , and fsa 1 hd.

A spherical balloon with radius r inches has volume Vsrd 4 3r 3 . Find a function that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r 1 1 inches.

2932 Evaluate the difference quotient for the given function. Simplify your answer. 29. fsxd 4 1 3x 2 x 2 , fs3 1 hd 2 fs3d h 30. f sxd x 3 , fsa 1 hd 2 fsad h 31. fsxd 1 x , fsxd 2 fsad x 2 a 32. fsxd x 1 3 x 1 1 , fsxd 2 fs1d x 2 1

2932 Evaluate the difference quotient for the given function. Simplify your answer. 29. fsxd 4 1 3x 2 x 2 , fs3 1 hd 2 fs3d h 30. f sxd x 3 , fsa 1 hd 2 fsad h 31. fsxd 1 x , fsxd 2 fsad x 2 a 32. fsxd x 1 3 x 1 1 , fsxd 2 fs1d x 2 2

2932 Evaluate the difference quotient for the given function. Simplify your answer. 29. fsxd 4 1 3x 2 x 2 , fs3 1 hd 2 fs3d h 30. f sxd x 3 , fsa 1 hd 2 fsad h 31. fsxd 1 x , fsxd 2 fsad x 2 a 32. fsxd x 1 3 x 1 1 , fsxd 2 fs1d x 2 3

2932 Evaluate the difference quotient for the given function. Simplify your answer. 29. fsxd 4 1 3x 2 x 2 , fs3 1 hd 2 fs3d h 30. f sxd x 3 , fsa 1 hd 2 fsad h 31. fsxd 1 x , fsxd 2 fsad x 2 a 32. fsxd x 1 3 x 1 1 , fsxd 2 fs1d x 2 4

3339 Find the domain of the function. 33. fsxd x 1 4 x 2 2 9 34. fsxd 2x 3 2 5 x 2 1 x 2 6 35. fstd s 3 2t 2 1 36. tstd s3 2 t 2 s2 1 t 37. hsxd 1 s 4 x 2 2 5x 38. fsud u 1 1 1 1 1 u 1 1 39. Fspd s2 2 sp

3339 Find the domain of the function. 33. fsxd x 1 4 x 2 2 9 34. fsxd 2x 3 2 5 x 2 1 x 2 6 35. fstd s 3 2t 2 1 36. tstd s3 2 t 2 s2 1 t 37. hsxd 1 s 4 x 2 2 5x 38. fsud u 1 1 1 1 1 u 1 1 39. Fspd s2 2 sp

3339 Find the domain of the function. 33. fsxd x 1 4 x 2 2 9 34. fsxd 2x 3 2 5 x 2 1 x 2 6 35. fstd s 3 2t 2 1 36. tstd s3 2 t 2 s2 1 t 37. hsxd 1 s 4 x 2 2 5x 38. fsud u 1 1 1 1 1 u 1 1 39. Fspd s2 2 sp

3339 Find the domain of the function. 33. fsxd x 1 4 x 2 2 9 34. fsxd 2x 3 2 5 x 2 1 x 2 6 35. fstd s 3 2t 2 1 36. tstd s3 2 t 2 s2 1 t 37. hsxd 1 s 4 x 2 2 5x 38. fsud u 1 1 1 1 1 u 1 1 39. Fspd s2 2 sp

3339 Find the domain of the function. 33. fsxd x 1 4 x 2 2 9 34. fsxd 2x 3 2 5 x 2 1 x 2 6 35. fstd s 3 2t 2 1 36. tstd s3 2 t 2 s2 1 t 37. hsxd 1 s 4 x 2 2 5x 38. fsud u 1 1 1 1 1 u 1 1 39. Fspd s2 2 sp

3339 Find the domain of the function. 33. fsxd x 1 4 x 2 2 9 34. fsxd 2x 3 2 5 x 2 1 x 2 6 35. fstd s 3 2t 2 1 36. tstd s3 2 t 2 s2 1 t 37. hsxd 1 s 4 x 2 2 5x 38. fsud u 1 1 1 1 1 u 1 1 39. Fspd s2 2 sp

3339 Find the domain of the function. 33. fsxd x 1 4 x 2 2 9 34. fsxd 2x 3 2 5 x 2 1 x 2 6 35. fstd s 3 2t 2 1 36. tstd s3 2 t 2 s2 1 t 37. hsxd 1 s 4 x 2 2 5x 38. fsud u 1 1 1 1 1 u 1 1 39. Fspd s2 2 sp

Find the domain and range and sketch the graph of the function hsxd s4 2 x 2 .

4152 Find the domain and sketch the graph of the function. 41. fsxd 2 2 0.4x 42. Fsxd x 2 2 2x 1 1 43. fstd 2t 1 t 2 44. Hstd 4 2 t 2 2 2 t 45. tsxd sx 2 5 46. Fsxd | 2x 1 1 | 47. Gsxd 3x 1 | x | x 48. tsxd | x | 2 x 49. fsxd H x 1 2 1 2 x if x , 0 if x > 0 50. fsxd H 3 2 1 2 x 2x 2 5 if x < 2 if x . 2 51. f sxd H x 1 2 x 2 if x < 21 if x . 21 52. f sxd H x 1 9 22x 26 if x , 23 if | x | < 3 if x . 3

4152 Find the domain and sketch the graph of the function. 41. fsxd 2 2 0.4x 42. Fsxd x 2 2 2x 1 1 43. fstd 2t 1 t 2 44. Hstd 4 2 t 2 2 2 t 45. tsxd sx 2 5 46. Fsxd | 2x 1 1 | 47. Gsxd 3x 1 | x | x 48. tsxd | x | 2 x 49. fsxd H x 1 2 1 2 x if x , 0 if x > 0 50. fsxd H 3 2 1 2 x 2x 2 5 if x < 2 if x . 2 51. f sxd H x 1 2 x 2 if x < 21 if x . 21 52. f sxd H x 1 9 22x 26 if x , 23 if | x | < 3 if x . 4

4152 Find the domain and sketch the graph of the function. 41. fsxd 2 2 0.4x 42. Fsxd x 2 2 2x 1 1 43. fstd 2t 1 t 2 44. Hstd 4 2 t 2 2 2 t 45. tsxd sx 2 5 46. Fsxd | 2x 1 1 | 47. Gsxd 3x 1 | x | x 48. tsxd | x | 2 x 49. fsxd H x 1 2 1 2 x if x , 0 if x > 0 50. fsxd H 3 2 1 2 x 2x 2 5 if x < 2 if x . 2 51. f sxd H x 1 2 x 2 if x < 21 if x . 21 52. f sxd H x 1 9 22x 26 if x , 23 if | x | < 3 if x . 5

4152 Find the domain and sketch the graph of the function. 41. fsxd 2 2 0.4x 42. Fsxd x 2 2 2x 1 1 43. fstd 2t 1 t 2 44. Hstd 4 2 t 2 2 2 t 45. tsxd sx 2 5 46. Fsxd | 2x 1 1 | 47. Gsxd 3x 1 | x | x 48. tsxd | x | 2 x 49. fsxd H x 1 2 1 2 x if x , 0 if x > 0 50. fsxd H 3 2 1 2 x 2x 2 5 if x < 2 if x . 2 51. f sxd H x 1 2 x 2 if x < 21 if x . 21 52. f sxd H x 1 9 22x 26 if x , 23 if | x | < 3 if x . 6

4152 Find the domain and sketch the graph of the function. 41. fsxd 2 2 0.4x 42. Fsxd x 2 2 2x 1 1 43. fstd 2t 1 t 2 44. Hstd 4 2 t 2 2 2 t 45. tsxd sx 2 5 46. Fsxd | 2x 1 1 | 47. Gsxd 3x 1 | x | x 48. tsxd | x | 2 x 49. fsxd H x 1 2 1 2 x if x , 0 if x > 0 50. fsxd H 3 2 1 2 x 2x 2 5 if x < 2 if x . 2 51. f sxd H x 1 2 x 2 if x < 21 if x . 21 52. f sxd H x 1 9 22x 26 if x , 23 if | x | < 3 if x . 7

4152 Find the domain and sketch the graph of the function. 41. fsxd 2 2 0.4x 42. Fsxd x 2 2 2x 1 1 43. fstd 2t 1 t 2 44. Hstd 4 2 t 2 2 2 t 45. tsxd sx 2 5 46. Fsxd | 2x 1 1 | 47. Gsxd 3x 1 | x | x 48. tsxd | x | 2 x 49. fsxd H x 1 2 1 2 x if x , 0 if x > 0 50. fsxd H 3 2 1 2 x 2x 2 5 if x < 2 if x . 2 51. f sxd H x 1 2 x 2 if x < 21 if x . 21 52. f sxd H x 1 9 22x 26 if x , 23 if | x | < 3 if x . 8

4152 Find the domain and sketch the graph of the function. 41. fsxd 2 2 0.4x 42. Fsxd x 2 2 2x 1 1 43. fstd 2t 1 t 2 44. Hstd 4 2 t 2 2 2 t 45. tsxd sx 2 5 46. Fsxd | 2x 1 1 | 47. Gsxd 3x 1 | x | x 48. tsxd | x | 2 x 49. fsxd H x 1 2 1 2 x if x , 0 if x > 0 50. fsxd H 3 2 1 2 x 2x 2 5 if x < 2 if x . 2 51. f sxd H x 1 2 x 2 if x < 21 if x . 21 52. f sxd H x 1 9 22x 26 if x , 23 if | x | < 3 if x . 9

4152 Find the domain and sketch the graph of the function. 41. fsxd 2 2 0.4x 42. Fsxd x 2 2 2x 1 1 43. fstd 2t 1 t 2 44. Hstd 4 2 t 2 2 2 t 45. tsxd sx 2 5 46. Fsxd | 2x 1 1 | 47. Gsxd 3x 1 | x | x 48. tsxd | x | 2 x 49. fsxd H x 1 2 1 2 x if x , 0 if x > 0 50. fsxd H 3 2 1 2 x 2x 2 5 if x < 2 if x . 2 51. f sxd H x 1 2 x 2 if x < 21 if x . 21 52. f sxd H x 1 9 22x 26 if x , 23 if | x | < 3 if x . 10

4152 Find the domain and sketch the graph of the function. 41. fsxd 2 2 0.4x 42. Fsxd x 2 2 2x 1 1 43. fstd 2t 1 t 2 44. Hstd 4 2 t 2 2 2 t 45. tsxd sx 2 5 46. Fsxd | 2x 1 1 | 47. Gsxd 3x 1 | x | x 48. tsxd | x | 2 x 49. fsxd H x 1 2 1 2 x if x , 0 if x > 0 50. fsxd H 3 2 1 2 x 2x 2 5 if x < 2 if x . 2 51. f sxd H x 1 2 x 2 if x < 21 if x . 21 52. f sxd H x 1 9 22x 26 if x , 23 if | x | < 3 if x . 11

4152 Find the domain and sketch the graph of the function. 41. fsxd 2 2 0.4x 42. Fsxd x 2 2 2x 1 1 43. fstd 2t 1 t 2 44. Hstd 4 2 t 2 2 2 t 45. tsxd sx 2 5 46. Fsxd | 2x 1 1 | 47. Gsxd 3x 1 | x | x 48. tsxd | x | 2 x 49. fsxd H x 1 2 1 2 x if x , 0 if x > 0 50. fsxd H 3 2 1 2 x 2x 2 5 if x < 2 if x . 2 51. f sxd H x 1 2 x 2 if x < 21 if x . 21 52. f sxd H x 1 9 22x 26 if x , 23 if | x | < 3 if x . 12

4152 Find the domain and sketch the graph of the function. 41. fsxd 2 2 0.4x 42. Fsxd x 2 2 2x 1 1 43. fstd 2t 1 t 2 44. Hstd 4 2 t 2 2 2 t 45. tsxd sx 2 5 46. Fsxd | 2x 1 1 | 47. Gsxd 3x 1 | x | x 48. tsxd | x | 2 x 49. fsxd H x 1 2 1 2 x if x , 0 if x > 0 50. fsxd H 3 2 1 2 x 2x 2 5 if x < 2 if x . 2 51. f sxd H x 1 2 x 2 if x < 21 if x . 21 52. f sxd H x 1 9 22x 26 if x , 23 if | x | < 3 if x . 13

4152 Find the domain and sketch the graph of the function. 41. fsxd 2 2 0.4x 42. Fsxd x 2 2 2x 1 1 43. fstd 2t 1 t 2 44. Hstd 4 2 t 2 2 2 t 45. tsxd sx 2 5 46. Fsxd | 2x 1 1 | 47. Gsxd 3x 1 | x | x 48. tsxd | x | 2 x 49. fsxd H x 1 2 1 2 x if x , 0 if x > 0 50. fsxd H 3 2 1 2 x 2x 2 5 if x < 2 if x . 2 51. f sxd H x 1 2 x 2 if x < 21 if x . 21 52. f sxd H x 1 9 22x 26 if x , 23 if | x | < 3 if x . 14

5357 Find a formula for the described function and state its domain. 53. A rectangle has perimeter 20 m. Express the area of the rectangle as a function of the length of one of its sides.

5357 Find a formula for the described function and state its domain. 54. A rectangle has area 16 m2 . Express the perimeter of the rectangle as a function of the length of one of its sides.

5357 Find a formula for the described function and state its domain. 55. Express the area of an equilateral triangle as a function of the length of a side.

5357 Find a formula for the described function and state its domain. 56. Express the surface area of a cube as a function of its volume.

5357 Find a formula for the described function and state its domain. 57. An open rectangular box with volume 2 m3 has a square base. Express the surface area of the box as a function of the length of a side of the base.

A cell phone plan has a basic charge of $35 a month. The plan includes 400 free minutes and charges 10 cents for each additional minute of usage. Write the monthly cost C as a function of the number x of minutes used and graph C as a function of x for 0 < x < 600.

A hotel chain charges $75 each night for the first two nights and $50 for each additional nights stay. Express the total cost T as a function of the number of nights x that a guest stays.

The function in Example 11 is called a step function because its graph looks like stairs. Give two other examples of step functions that arise in everyday life.

Temperature chart The figure shows the temperature of a patient infected with the malaria species P. malariae. Estimate the period of the temp

A temperature chart is shown for a patient with a fever induced by the malaria species P. falciparum. What do you think is happening

Graphs of f and t are shown. Decide whether each function is even, odd, or neither. Explain your reasoning

Graphs of f and t are shown. Decide whether each function is even, odd, or neither. Explain your reasoning

a) If the point s5, 3d is on the graph of an even function, what other point must also be on the graph? (b) If the point s5, 3d is on the graph of an odd function, what other point must also be on the graph?

A function f has domain f25, 5g and a portion of its graph is shown. (a) Complete the graph of f if it is known that f is even. (b) Complete the graph of f if it is known that f is odd

6772 Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. 67. fsxd x x 2 1 1 68. fsxd x 2 x 4 1 1 69. fsxd x x 1 1 70. fsxd x | x | 71. fsxd 1 1 3x 2 2 x 4 72. fsxd 1 1 3x 3 2 x 5

6772 Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. 67. fsxd x x 2 1 1 68. fsxd x 2 x 4 1 1 69. fsxd x x 1 1 70. fsxd x | x | 71. fsxd 1 1 3x 2 2 x 4 72. fsxd 1 1 3x 3 2 x 6

6772 Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. 67. fsxd x x 2 1 1 68. fsxd x 2 x 4 1 1 69. fsxd x x 1 1 70. fsxd x | x | 71. fsxd 1 1 3x 2 2 x 4 72. fsxd 1 1 3x 3 2 x 7

6772 Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. 67. fsxd x x 2 1 1 68. fsxd x 2 x 4 1 1 69. fsxd x x 1 1 70. fsxd x | x | 71. fsxd 1 1 3x 2 2 x 4 72. fsxd 1 1 3x 3 2 x 8

6772 Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. 67. fsxd x x 2 1 1 68. fsxd x 2 x 4 1 1 69. fsxd x x 1 1 70. fsxd x | x | 71. fsxd 1 1 3x 2 2 x 4 72. fsxd 1 1 3x 3 2 x 9

6772 Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. 67. fsxd x x 2 1 1 68. fsxd x 2 x 4 1 1 69. fsxd x x 1 1 70. fsxd x | x | 71. fsxd 1 1 3x 2 2 x 4 72. fsxd 1 1 3x 3 2 x 10

If f and t are both even functions, is f 1 t even? If f and t are both odd functions, is f 1 t odd? What if f is even and t is odd? Justify your answers.

If f and t are both even functions, is the product ft even? If f and t are both odd functions, is ft odd? What if f is even and t is odd? Justify your answers.