Chapter 1: Functions - OpenTextBookStore



Section 1.1 Exercises

1. The amount of garbage, G, produced by a city with population p is given by [pic]. G is measured in tons per week, and p is measured in thousands of people.

a. The town of Tola has a population of 40,000 and produces 13 tons of garbage each week. Express this information in terms of the function f.

b. Explain the meaning of the statement [pic]

2. The number of cubic yards of dirt, D, needed to cover a garden with area a square feet is given by [pic].

a. A garden with area 5000 ft2 requires 50 cubic yards of dirt. Express this information in terms of the function g.

b. Explain the meaning of the statement [pic]

3. Let [pic] be the number of ducks in a lake t years after 1990. Explain the meaning of each statement:

a. [pic] b. [pic]

4. Let [pic] be the height above ground, in feet, of a rocket t seconds after launching. Explain the meaning of each statement:

a. [pic] b. [pic]

5. Select all of the following graphs which represent y as a function of x.

a[pic] b [pic] c[pic]

d [pic] e [pic] f[pic]

 

6. Select all of the following graphs which represent y as a function of x.

a[pic] b[pic] c[pic]

d[pic] e [pic] f[pic]

 

7. Select all of the following tables which represent y as a function of x.

|a. |x |b. |x |c. |x |

| |5 | |5 | |5 |

| |10 | |10 | |10 |

| |15 | |15 | |10 |

| | | | | | |

| |y | |y | |y |

| |3 | |3 | |3 |

| |8 | |8 | |8 |

| |14 | |8 | |14 |

| | | | | | |

8. Select all of the following tables which represent y as a function of x.

|a. |x |b. |x |c. |x |

| |2 | |2 | |2 |

| |6 | |6 | |6 |

| |13 | |6 | |13 |

| | | | | | |

| |y | |y | |y |

| |3 | |3 | |3 |

| |10 | |10 | |10 |

| |10 | |14 | |14 |

| | | | | | |

9. Select all of the following tables which represent y as a function of x.

|a. |x |b. |x |c. |x |d. |x |

| |y | |y | |y | |y |

| | | | | | | | |

| |0 | |-1 | |0 | |-1 |

| |-2 | |-4 | |-5 | |-4 |

| | | | | | | | |

| |3 | |2 | |3 | |1 |

| |1 | |3 | |1 | |2 |

| | | | | | | | |

| |4 | |5 | |3 | |4 |

| |6 | |4 | |4 | |2 |

| | | | | | | | |

| |8 | |8 | |9 | |9 |

| |9 | |7 | |8 | |7 |

| | | | | | | | |

| |3 | |12 | |16 | |12 |

| |1 | |11 | |13 | |13 |

| | | | | | | | |

| | | | | | | | |

10. Select all of the following tables which represent y as a function of x.

|a. |x |b. |x |c. |x |d. |x |

| |y | |y | |y | |y |

| | | | | | | | |

| |-4 | |-5 | |-1 | |-1 |

| |-2 | |-3 | |-3 | |-5 |

| | | | | | | | |

| |3 | |2 | |1 | |3 |

| |2 | |1 | |2 | |1 |

| | | | | | | | |

| |6 | |2 | |5 | |5 |

| |4 | |4 | |4 | |1 |

| | | | | | | | |

| |9 | |7 | |9 | |8 |

| |7 | |9 | |8 | |7 |

| | | | | | | | |

| |12 | |11 | |1 | |14 |

| |16 | |10 | |2 | |12 |

| | | | | | | | |

11. Select all of the following tables which represent y as a function of x and are one-to-one.

|a. |x |b. |x |c. |x |

| |3 | |3 | |3 |

| |8 | |8 | |8 |

| |12 | |12 | |8 |

| | | | | | |

| |y | |y | |y |

| |4 | |4 | |4 |

| |7 | |7 | |7 |

| |7 | |13 | |13 |

| | | | | | |

12. Select all of the following tables which represent y as a function of x and are one-to-one.

|a. |x |b. |x |c. |x |

| |2 | |2 | |2 |

| |8 | |8 | |8 |

| |8 | |14 | |14 |

| | | | | | |

| |y | |y | |y |

| |5 | |5 | |5 |

| |6 | |6 | |6 |

| |13 | |6 | |13 |

| | | | | | |

13. Select all of the following graphs which are one-to-one functions.

a.[pic] b.[pic] c.[pic]

d.[pic] e. [pic] f.[pic]

 

14. Select all of the following graphs which are one-to-one functions.

a[pic] b[pic] c[pic]

d[pic] e [pic] f [pic]

 

Given the each function [pic] graphed, evaluate [pic] and [pic]

15.[pic] 16.[pic]

|Given the function [pic] graphed here, |Given the function [pic] graphed here. |

|Evaluate [pic] |Evaluate [pic] |

|Solve [pic] |Solve [pic] |

|[pic] |[pic] |

19. Based on the table below,

a. Evaluate [pic] b. Solve [pic]

|x |0 |1 |2 |3 |4 |5 |6 |

| | | | | | | | |

|29. |x |30. |x |31. |x |32. |x |

| |f(x) | |g(x) | |h(x) | |k(x) |

| | | | | | | | |

| |1 | |1 | |1 | |1 |

| |-10 | |-200 | |-100 | |-50 |

| | | | | | | | |

| |2 | |2 | |2 | |2 |

| |-25 | |-190 | |-50 | |-100 |

| | | | | | | | |

| |3 | |3 | |3 | |3 |

| |-37 | |-160 | |-25 | |-200 |

| | | | | | | | |

| |4 | |4 | |4 | |4 |

| |-47 | |-100 | |-10 | |-400 |

| | | | | | | | |

| |5 | |5 | |5 | |5 |

| |-54 | |0 | |0 | |-900 |

| | | | | | | | |

| | | | | | | | |

For each function graphed, estimate the intervals on which the function is concave up and concave down, and the location of any inflection points.

33. [pic] 34. [pic]

35. [pic] 36. [pic]

Use a graph to estimate the local extrema and inflection points of each function, and to estimate the intervals on which the function is increasing, decreasing, concave up, and concave down.

37. [pic] 38. [pic]

39. [pic] 40. [pic]

41. [pic] 42. [pic]

Section 1.4 Exercises

Given each pair of equations, calculate [pic] and [pic]

1. [pic], [pic] 2. [pic], [pic]

3. [pic], [pic] 4. [pic], [pic]

Use the table of values to evaluate each expression

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic]

11. [pic]

12. [pic]

Use the graphs to evaluate the expressions below.

13. [pic]

14. [pic]

15. [pic]

16. [pic]

17. [pic]

18. [pic]

19. [pic]

20. [pic]

For each pair of functions, find [pic] and [pic]. Simplify your answers.

21. [pic], [pic] 22. [pic], [pic]

23. [pic], [pic] 24. [pic], [pic]

25. [pic], [pic] 26. [pic] , [pic]

27. If [pic],[pic]and [pic], find [pic] 

28. If [pic], [pic] and [pic] , find [pic] 

29. Given functions [pic] and [pic], state the domains of the following functions using interval notation.

a. Domain of [pic]

b. Domain of [pic]

c. Domain of [pic]

30. Given functions [pic] and [pic], state the domains of the following functions using interval notation.

a. Domain of [pic]

b. Domain of [pic]

c. Domain of [pic]

31. The function [pic] gives the number of items that will be demanded when the price is p. The production cost, [pic] is the cost of producing x items. To determine the cost of production when the price is $6, you would do which of the following:

a. Evaluate [pic] b. Evaluate [pic]

c. Solve [pic] d. Solve [pic]

32. The function [pic] gives the pain level on a scale of 0-10 experienced by a patient with d milligrams of a pain reduction drug in their system. The milligrams of drug in the patient’s system after t minutes is modeled by [pic]. To determine when the patient will be at a pain level of 4, you would need to:

a. Evaluate [pic] b. Evaluate [pic]

c. Solve [pic] d. Solve [pic]

33. The radius r, in inches, of a balloon is related to the volume, V, by [pic]. Air is pumped into the balloon, so the volume after t seconds is given by [pic]

a. Find the composite function [pic]

b. Find the time when the radius reaches 10 inches.

34. The number of bacteria in a refrigerated food product is given by [pic], [pic] where T is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by [pic], where t is the time in hours.

a. Find the composite function [pic]

b. Find the time when the bacteria count reaches 6752

Find functions [pic] and [pic] so the given function can be expressed as [pic]

35. [pic] 36. [pic]

37. [pic] 38. [pic]

39. [pic] 40. [pic]

41. Let [pic] be a linear function, having form [pic] for constants a and b. [UW]

a. Show that [pic] is a linear function

b. Find a function [pic] such that [pic]

42. Let [pic] [UW]

a. Sketch the graphs of [pic] on the interval −2 ≤ x ≤ 10.

b. Your graphs should all intersect at the point (6, 6). The value x = 6 is called a fixed point of the function f(x)since [pic]; that is, 6 is fixed - it doesn’t move when f is applied to it. Give an explanation for why 6 is a fixed point for any function [pic].

c. Linear functions (with the exception of [pic]) can have at most one fixed point. Quadratic functions can have at most two. Find the fixed points of the function [pic].

d. Give a quadratic function whose fixed points are x = −2 and x = 3.

43. A car leaves Seattle heading east. The speed of the car in mph after m minutes is given by the function [pic]. [UW]

a. Find a function [pic] that converts seconds s into minutes m. Write out the formula for the new function [pic]; what does this function calculate?

b. Find a function [pic]) that converts hours h into minutes m. Write out the formula for the new function [pic]; what does this function calculate?

c. Find a function [pic] that converts mph s into ft/sec z. Write out the formula for the new function [pic]; what does this function calculate?

Section 1.5 Exercises

Describe how each function is a transformation of the original function [pic]

1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic] 6. [pic]

7. [pic] 8. [pic]

9. [pic] 10. [pic]

11. Write a formula for [pic] shifted up 1 unit and left 2 units

12. Write a formula for [pic] shifted down 3 units and right 1 unit

13. Write a formula for [pic] shifted down 4 units and right 3 units

14. Write a formula for [pic] shifted up 2 units and left 4 units

15. Tables of values for [pic], [pic], and [pic] are given below. Write [pic] and [pic] as transformations of [pic].

|x |x |x |

|-2 |-1 |-2 |

|-1 |0 |-1 |

|0 |1 |0 |

|1 |2 |1 |

|2 |3 |2 |

| | | |

|f(x) |g(x) |h(x) |

|-2 |-2 |-1 |

|-1 |-1 |0 |

|-3 |-3 |-2 |

|1 |1 |2 |

|2 |2 |3 |

| | | |

16. Tables of values for [pic], [pic], and [pic] are given below. Write [pic] and [pic] as transformations of [pic].

|x |x |x |

|-2 |-3 |-2 |

|-1 |-2 |-1 |

|0 |-1 |0 |

|1 |0 |1 |

|2 |1 |2 |

| | | |

|f(x) |g(x) |h(x) |

|-1 |-1 |-2 |

|-3 |-3 |-4 |

|4 |4 |3 |

|2 |2 |1 |

|1 |1 |0 |

| | | |

The graph of [pic] is shown. Sketch a graph of each transformation of [pic]

17. [pic]

18. [pic]

19. [pic]

20. [pic]

Sketch a graph of each function as a transformation of a toolkit function

21. [pic]

22. [pic]

23. [pic]

24. [pic]

 Write an equation for the function graphed below

25.[pic] 26.[pic]

27.[pic] 28.[pic]

Find a formula for each of the transformations of the square root whose graphs are given below.

29. [pic] 30. [pic]

 

 

The graph of [pic] is shown. Sketch a graph of each transformation of [pic]

31. [pic]

32. [pic]

  

33. Starting with the graph of [pic] write the equation of the graph that results from

a. reflecting [pic] about the x-axis and the y-axis

b. reflecting [pic] about the x-axis, shifting left 2 units, and down 3 units

 

34. Starting with the graph of [pic] write the equation of the graph that results from

a. reflecting [pic] about the x-axis

b. reflecting [pic] about the y-axis, shifting right 4 units, and up 2 units

Write an equation for the function graphed below

35. [pic] 36. [pic]

37. [pic] 38. [pic]

39. For each equation below, determine if the function is Odd, Even, or Neither

a. [pic]

b. [pic]

c. [pic]

 

40. For each equation below, determine if the function is Odd, Even, or Neither

a. [pic]

b. [pic]

c. [pic]

 

Describe how each function is a transformation of the original function [pic]

41. [pic] 42. [pic]

43. [pic] 44. [pic]

45. [pic] 46. [pic]

47. [pic] 48. [pic]

49. [pic] 50. [pic]

51. Write a formula for [pic] reflected over the y axis and horizontally compressed by a factor of [pic]

52. Write a formula for [pic] reflected over the x axis and horizontally stretched by a factor of 2

53. Write a formula for [pic] vertically compressed by a factor of [pic], then shifted to the left 2 units and down 3 units.

54. Write a formula for [pic] vertically stretched by a factor of 8, then shifted to the right 4 units and up 2 units.

55. Write a formula for [pic] horizontally compressed by a factor of [pic], then shifted to the right 5 units and up 1 unit.

56. Write a formula for [pic] horizontally stretched by a factor of 3, then shifted to the left 4 units and down 3 units.

Describe how each formula is a transformation of a toolkit function. Then sketch a graph of the transformation.

57. [pic] 58. [pic]

59. [pic] 60. [pic]

61. [pic] 62. [pic]

63. [pic] 64. [pic]

65. [pic] 66. [pic]

Determine the interval(s) on which the function is increasing and decreasing

67. [pic] 68. [pic]

69. [pic] 70. [pic]

Determine the interval(s) on which the function is concave up and concave down

71. [pic] 72. [pic]

73. [pic] 74. [pic]

The function [pic] is graphed here. Write an equation for each graph below as a transformation of [pic].

75.[pic] 76.[pic] 77.[pic]

78.[pic] 79.[pic] 80.[pic]

81.[pic] 82.[pic] 83.[pic]

84.[pic] 85.[pic] 86.[pic]

Write an equation for the transformed toolkit function graphed below.

87.[pic] 88.[pic] 89.[pic]

90.[pic] 91.[pic] 92.[pic]

93.[pic] 94.[pic] 95.[pic]

96.[pic] 97.[pic] 98. [pic]

99. Suppose you have a function [pic] such that the domain of [pic] is 1 ≤ x ≤ 6 and the range of [pic] is −3 ≤ y ≤ 5. [UW]

a. What is the domain of[pic]?

b. What is the range of [pic] ?

c. What is the domain of [pic] ?

d. What is the range of [pic] ?

e. Can you find constants B and C so that the domain of [pic] is 8 ≤ x ≤ 9?

f. Can you find constants A and D so that the range of [pic] is 0 ≤ y ≤ 1?

Section 1.6 Exercises

Assume that the function f is a one-to-one function.

1. If [pic] , find [pic] 2. If [pic] , find [pic]

3. If [pic], find [pic] 4. If [pic], find [pic]

5. If [pic], find [pic] 6. If [pic], find [pic]

7. Using the graph of [pic] shown

a. Find [pic]

b. Solve [pic]

c. Find [pic]

d. Solve [pic]

 

8. Using the graph shown

a. Find [pic]

b. Solve [pic]

c. Find [pic]

d. Solve [pic]

9. Use the table below to fill in the missing values.

For each function below, find [pic] 

13. [pic] 14. [pic]

15. [pic] 16. [pic]

17. [pic] 18. [pic]

For each function, find a domain on which f is one-to-one and non-decreasing, then find the inverse of f restricted to that domain.

19. [pic] 20. [pic]

21. [pic]  22. [pic]

23. If [pic] and [pic], find

a. [pic]

b. [pic]

c. What does this tell us about the relationship between [pic] and [pic]?

24. If [pic] and [pic], find

a. [pic]

b. [pic]

c. What does this tell us about the relationship between [pic] and [pic]?

-----------------------

x

f(x)

a

b

c

p

r

t

K

L

|x | [pic] |[pic] |

|0 |7 |9 |

|1 |6 |5 |

|2 |5 |6 |

|3 |8 |2 |

|4 |4 |1 |

|5 |0 |8 |

|6 |2 |7 |

|7 |1 |3 |

|8 |9 |4 |

|9 |3 |0 |

[pic]

i. Cube root

ii. Reciprocal

iii. Linear

iv. Square Root

v. Absolute Value

vi. Quadratic

vii. Reciprocal Squared

viii. Cubic

|i. |ii. |iii. |iv. |

|[pic] |[pic] |[pic] |[pic] |

| | | | |

|v. |vi. |vii. |viii. |

|[pic] |[pic] |[pic] |[pic] |

|i. |In |ii. |In |iii. |In |

| |Out | |Out | |Out |

| | | | | | |

| |-2 | |-2 | |-2 |

| |-0.5 | |-2 | |-8 |

| | | | | | |

| |-1 | |-1 | |-1 |

| |-1 | |-1 | |-1 |

| | | | | | |

| |0 | |0 | |0 |

| |_ | |0 | |0 |

| | | | | | |

| |1 | |1 | |1 |

| |1 | |1 | |1 |

| | | | | | |

| |2 | |2 | |2 |

| |0.5 | |2 | |8 |

| | | | | | |

| |3 | |3 | |3 |

| |0.33 | |3 | |27 |

| | | | | | |

| | | | | | |

|iv. |In |v. |In |vi. |In |

| |Out | |Out | |Out |

| | | | | | |

| |-2 | |-2 | |-2 |

| |4 | |_ | |2 |

| | | | | | |

| |-1 | |-1 | |-1 |

| |1 | |_ | |1 |

| | | | | | |

| |0 | |0 | |0 |

| |0 | |0 | |0 |

| | | | | | |

| |1 | |1 | |1 |

| |1 | |1 | |1 |

| | | | | | |

| |2 | |4 | |2 |

| |4 | |2 | |2 |

| | | | | | |

| |3 | |9 | |3 |

| |9 | |3 | |3 |

| | | | | | |

|i. |In |ii. |In |iii. |In |

| |Out | |Out | |Out |

| | | | | | |

| |-2 | |-2 | |-2 |

| |-0.5 | |-2 | |-8 |

| | | | | | |

| |-1 | |-1 | |-1 |

| |-1 | |-1 | |-1 |

| | | | | | |

| |0 | |0 | |0 |

| |_ | |0 | |0 |

| | | | | | |

| |1 | |1 | |1 |

| |1 | |1 | |1 |

| | | | | | |

| |2 | |2 | |2 |

| |0.5 | |2 | |8 |

| | | | | | |

| |3 | |3 | |3 |

| |0.33 | |3 | |27 |

| | | | | | |

| | | | | | |

|iv. |In |v. |In |vi. |In |

| |Out | |Out | |Out |

| | | | | | |

| |-2 | |-2 | |-2 |

| |4 | |_ | |2 |

| | | | | | |

| |-1 | |-1 | |-1 |

| |1 | |_ | |1 |

| | | | | | |

| |0 | |0 | |0 |

| |0 | |0 | |0 |

| | | | | | |

| |1 | |1 | |1 |

| |1 | |1 | |1 |

| | | | | | |

| |2 | |4 | |2 |

| |4 | |2 | |2 |

| | | | | | |

| |3 | |9 | |3 |

| |9 | |3 | |3 |

| | | | | | |

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