3.9.0: Review Exercises

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Review Exercises

Functions and Function Notation

For the following exercises, determine whether the relation is a function.

${(a, b), (c, d), (e, d)}$

${(5, 2), (6, 1), (6, 2), (4, 8)}$

$y^{2} + 4 = x,$ for $x$ the independent variable and $y$ the dependent variable

Is the graph in Figure 1 a function?

Figure 1

For the following exercises, evaluate the function at the indicated values: $f (- 3); f (2); f (- a); - f (a); f (a + h) .$

$f (x) = - 2 x^{2} + 3 x$

$f (x) = 2 | 3 x - 1 |$

For the following exercises, determine whether the functions are one-to-one.

$f (x) = - 3 x + 5$

$f (x) = | x - 3 |$

For the following exercises, use the vertical line test to determine if the relation whose graph is provided is a function.

$Graph of a cubic function.$

10.

$Graph of a relation.$

11.

$Graph of a relation.$

For the following exercises, graph the functions.

12.

$f (x) = | x + 1 |$

13.

$f (x) = x^{2} - 2$

For the following exercises, use Figure 2 to approximate the values.

Figure 2

14.

$f (2)$

15.

$f (−2)$

16.

If $f (x) = −2,$ then solve for $x .$

17.

If $f (x) = 1,$ then solve for $x .$

For the following exercises, use the function $h (t) = - 16 t^{2} + 80 t$ to find the values.

18.

$\frac{h (2) - h (1)}{2 - 1}$

19.

$\frac{h (a) - h (1)}{a - 1}$

Domain and Range

For the following exercises, find the domain of each function, expressing answers using interval notation.

20.

$f (x) = \frac{2}{3 x + 2}$

21.

$f (x) = \frac{x - 3}{x^{2} - 4 x - 12}$

22.

$f (x) = \frac{\sqrt{x - 6}}{\sqrt{x - 4}}$

23.

Graph this piecewise function: $f (x) = {\begin{cases} x + 1 x < - 2 \\ - 2 x - 3 x \geq - 2 \end{cases}$

Rates of Change and Behavior of Graphs

For the following exercises, find the average rate of change of the functions from $x = 1 to x = 2.$

24.

$f (x) = 4 x - 3$

25.

$f (x) = 10 x^{2} + x$

26.

$f (x) = - \frac{2}{x^{2}}$

For the following exercises, use the graphs to determine the intervals on which the functions are increasing, decreasing, or constant.

27.

$Graph of a parabola.$

28.

$Graph of a cubic function.$

29.

$Graph of a function.$

30.

Find the local minimum of the function graphed in Exercise 1.27.

31.

Find the local extrema for the function graphed in Exercise 1.28.

32.

For the graph in Figure 3, the domain of the function is $[- 3, 3] .$ The range is $[- 10, 10] .$ Find the absolute minimum of the function on this interval.

33.

Find the absolute maximum of the function graphed in Figure 3.

Figure 3

Composition of Functions

For the following exercises, find $(f \circ g) (x)$ and $(g \circ f) (x)$ for each pair of functions.

34.

$f (x) = 4 - x, g (x) = - 4 x$

35.

$f (x) = 3 x + 2, g (x) = 5 - 6 x$

36.

$f (x) = x^{2} + 2 x, g (x) = 5 x + 1$

37.

$f (x) = \sqrt{x + 2}, g (x) = \frac{1}{x}$

38.

$f (x) = \frac{x + 3}{2}, g (x) = \sqrt{1 - x}$

For the following exercises, find $(f \circ g)$ and the domain for $(f \circ g) (x)$ for each pair of functions.

39.

$f (x) = \frac{x + 1}{x + 4}, g (x) = \frac{1}{x}$

40.

$f (x) = \frac{1}{x + 3}, g (x) = \frac{1}{x - 9}$

41.

$f (x) = \frac{1}{x}, g (x) = \sqrt{x}$

42.

$f (x) = \frac{1}{x^{2} - 1}, g (x) = \sqrt{x + 1}$

For the following exercises, express each function $H$ as a composition of two functions $f$ and $g$ where $H (x) = (f \circ g) (x) .$

43.

$H (x) = \sqrt{\frac{2 x - 1}{3 x + 4}}$

44.

$H (x) = \frac{1}{{(3 x^{2} - 4)}^{- 3}}$

Transformation of Functions

For the following exercises, sketch a graph of the given function.

45.

$f (x) = {(x - 3)}^{2}$

46.

$f (x) = {(x + 4)}^{3}$

47.

$f (x) = \sqrt{x} + 5$

48.

$f (x) = - x^{3}$

49.

$f (x) = \sqrt[3]{- x}$

50.

$f (x) = 5 \sqrt{- x} - 4$

51.

$f (x) = 4 [| x - 2 | - 6]$

52.

$f (x) = - {(x + 2)}^{2} - 1$

For the following exercises, sketch the graph of the function $g$ if the graph of the function $f$ is shown in Figure 4.

Figure 4

53.

$g (x) = f (x - 1)$

54.

$g (x) = 3 f (x)$

For the following exercises, write the equation for the standard function represented by each of the graphs below.

55.

$Graph of an absolute function.$

56.

$Graph of a half circle.$

For the following exercises, determine whether each function below is even, odd, or neither.

57.

$f (x) = 3 x^{4}$

58.

$g (x) = \sqrt{x}$

59.

$h (x) = \frac{1}{x} + 3 x$

For the following exercises, analyze the graph and determine whether the graphed function is even, odd, or neither.

60.

$Graph of a parabola.$

61.

$Graph of a parabola.$

62.

$Graph of a cubic function.$

Absolute Value Functions

For the following exercises, write an equation for the transformation of $f (x) = | x | .$

63.

$Graph of f(x).$

64.

$Graph of f(x).$

65.

$Graph of f(x).$

For the following exercises, graph the absolute value function.

66.

$f (x) = | x - 5 |$

67.

$f (x) = - | x - 3 |$

68.

$f (x) = | 2 x - 4 |$

For the following exercises, solve the absolute value equation.

69.

$| x + 4 | = 18$

70.

$| \frac{1}{3} x + 5 | = | \frac{3}{4} x - 2 |$

For the following exercises, solve the inequality and express the solution using interval notation.

71.

$| 3 x - 2 | < 7$

72.

$| \frac{1}{3} x - 2 | \leq 7$

Inverse Functions

For the following exercises, find $f^{- 1} (x)$ for each function.

73.

$f (x) = 9 + 10 x$

74.

$f (x) = \frac{x}{x + 2}$

For the following exercise, find a domain on which the function $f$ is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of $f$ restricted to that domain.

75.

$f (x) = x^{2} + 1$

76.

Given $f (x) = x^{3} - 5$ and $g (x) = \sqrt[3]{x + 5} :$

Find $f (g (x))$ and $g (f (x)) .$
What does the answer tell us about the relationship between $f (x)$ and $g (x) ?$

For the following exercises, use a graphing utility to determine whether each function is one-to-one.

77.

$f (x) = \frac{1}{x}$

78.

$f (x) = - 3 x^{2} + x$

79.

If $f (5) = 2,$ find $f^{- 1} (2) .$

80.

If $f (1) = 4,$ find $f^{- 1} (4) .$

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Review Exercises

Functions and Function Notation

Domain and Range

Rates of Change and Behavior of Graphs

Composition of Functions

Transformation of Functions

Absolute Value Functions

Inverse Functions