Introduction to Functions

For the following exercises, determine whether the relation represents [latex]y[/latex] as a function of [latex]x[/latex].

1. [latex]y = x^2[/latex]
2. [latex]3 - \sqrt{6-2x}[/latex]
3. [latex]3x^2 + y = 14[/latex]
4. [latex]y = -2x^2 + 40x[/latex]
5. [latex]x = \dfrac{3y + 5}{7y - 1}[/latex]
6. [latex]y = \dfrac{3x + 5}{7x - 1}[/latex]
7. [latex]2xy = 1[/latex]
8. [latex]y = x^3[/latex]
9. [latex]x = \pm \sqrt{1 - y}[/latex]
10. [latex]y^2 = x^2[/latex]

For the following exercises, evaluate [latex]f(-3)[/latex], [latex]f(2)[/latex], [latex]f(-a)[/latex], [latex]-f(a)[/latex], [latex]f(a + h)[/latex].

11. [latex]f(x) = 2x - 5[/latex]
12. [latex]f(x) = \sqrt{2 - x + 5}[/latex]
13. [latex]f(x) = |x - 1| - |x + 1|[/latex]
14. Given the function [latex]g(x) = x^2 + 2x[/latex], evaluate [latex]\dfrac{g(x) - g(a)}{x - a}[/latex], [latex]x \neq a[/latex].
15. Given the function [latex]f(x) = 8 - 3x[/latex]:
    1. Evaluate [latex]f(-2)[/latex].
    2. Solve [latex]f(x) = -1[/latex].
16. Given the function [latex]f(x) = x^2 - 3x[/latex]:
    1. Evaluate [latex]f(5)[/latex].
    2. Solve [latex]f(x) = 4[/latex].

For the following exercises, use the vertical line test to determine which graphs show relations that are functions.

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27. Given the following graph,
    
    1. Evaluate [latex]f(0)[/latex].
    2. Solve for [latex]f(x) = -3[/latex].

For the following exercises, determine if the given graph is a one-to-one function.

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For the following exercise, determine whether the relation represents a function.

33. [latex]\{(3, 4), (4, 5), (5, 6)\}[/latex]

For the following exercises, determine if the relation represented in table form represents [latex]y[/latex] as a function of [latex]x[/latex].

34. [latex]x[/latex]	[latex]5[/latex]	[latex]10[/latex]	[latex]15[/latex]
    [latex]y[/latex]	[latex]3[/latex]	[latex]8[/latex]	[latex]14[/latex]

35. [latex]x[/latex]	[latex]5[/latex]	[latex]10[/latex]	[latex]10[/latex]
    [latex]y[/latex]	[latex]3[/latex]	[latex]8[/latex]	[latex]14[/latex]

For the following exercises, use the function [latex]f[/latex] represented in the table below.

[latex]x[/latex]	[latex]0[/latex]	[latex]1[/latex]	[latex]2[/latex]	[latex]3[/latex]	[latex]4[/latex]	[latex]5[/latex]	[latex]6[/latex]	[latex]7[/latex]	[latex]8[/latex]	[latex]9[/latex]
[latex]f(x)[/latex]	[latex]74[/latex]	[latex]28[/latex]	[latex]1[/latex]	[latex]53[/latex]	[latex]56[/latex]	[latex]3[/latex]	[latex]36[/latex]	[latex]45[/latex]	[latex]14[/latex]	[latex]47[/latex]

36. Solve [latex]f(x) = 1[/latex].

37. The number of cubic yards of dirt, [latex]D[/latex], needed to cover a garden with area [latex]a[/latex] square feet is given by [latex]D = g(a)[/latex].
    1. A garden with area [latex]5000 \, \text{ft}^2[/latex] requires [latex]50 \, \text{yd}^3[/latex] of dirt. Express this information in terms of the function [latex]g[/latex].
    2. Explain the meaning of the statement [latex]g(100) = 1[/latex].
38. Let [latex]h(t)[/latex] be the height above ground, in feet, of a rocket [latex]t[/latex] seconds after launching. Explain the meaning of each statement:
    1. [latex]h(1) = 200[/latex]
    2. [latex]h(2) = 350[/latex]

Domain and Range

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

1. [latex]f(x) = 5 - 2x^2[/latex]
2. [latex]f(x) = \sqrt{x^2 + 4}[/latex]
3. [latex]f(x) = \sqrt[3]{x-1}[/latex]
4. [latex]f(x) = \dfrac{3x + 1}{4x + 2}[/latex]
5. [latex]f(x) = \dfrac{x - 3}{x^2 + 9x - 22}[/latex]
6. [latex]f(x) = \dfrac{2x^3 - 250}{x^2 - 2x - 15}[/latex]
7. [latex]f(x) = \dfrac{2x + 1}{\sqrt{5 - x}}[/latex]
8. [latex]f(x) = \dfrac{\sqrt{x - 6}}{\sqrt{x - 4}}[/latex]
9. [latex]f(x) = \dfrac{x^2 - 9x}{x^2 - 81}[/latex]

For the following exercises, write the domain and range of each function using interval notation.

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For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.

16. [latex]f(x) = \begin{cases} 2x - 1 & \text{if } x < 1 \\ 1 + x & \text{if } x \geq 1 \end{cases}[/latex]
17. [latex]f(x) = \begin{cases} 3 & \text{if } x < 0 \\ \sqrt{x} & \text{if } x \geq 0 \end{cases}[/latex]
18. [latex]f(x) = \begin{cases} x^2 & \text{if } x < 0 \\ x + 2 & \text{if } x \geq 0 \end{cases}[/latex]
19. [latex]f(x) = \begin{cases} |x| & \text{if } x < 2 \\ 1 & \text{if } x \geq 2 \end{cases}[/latex]

For the following exercise, given each function [latex]f[/latex], evaluate [latex]f(-3)[/latex], [latex]f(-2)[/latex], [latex]f(-1)[/latex], and [latex]f(0)[/latex].

20. [latex]f(x) = \begin{cases} 1 & \text{if } x \leq -3 \\ 0 & \text{if } x > -3 \end{cases}[/latex]

For the following exercises, given each function [latex]f[/latex], evaluate [latex]f(-1)[/latex], [latex]f(0)[/latex], [latex]f(2)[/latex], and [latex]f(4)[/latex].

21. [latex]f(x) = \begin{cases} 7x + 3 & \text{if } x < 0 \\ 7x + 6 & \text{if } x \geq 0 \end{cases}[/latex]
22. [latex]f(x) = \begin{cases} 5x & \text{if } x < 0 \\ 3 & \text{if } 0 \leq x \leq 3 \\ x^2 & \text{if } x > 3 \end{cases}[/latex]

For the following exercise, write the domain for the piecewise function in interval notation.

23. [latex]f(x) = \begin{cases} x^2 - 2 & \text{if } x < 1 \\ -x^2 + 2 & \text{if } x \geq 1 \end{cases}[/latex]

Rates of Change and Behavior of Graphs

For the following exercises, find the average rate of change of each function on the interval specified for real numbers [latex]b[/latex] or [latex]h[/latex] in simplest form.

1. [latex]f(x) = 2x^2 + 1[/latex] on [latex][x, x + h][/latex]
2. [latex]a(t) = \dfrac{1}{t + 4}[/latex] on [latex][9, 9 + h][/latex]
3. [latex]j(x) = 3x^3[/latex] on [latex][1, 1 + h][/latex]
4. Find [latex]\dfrac{f(x+h)-f(x)}{h}[/latex] given [latex]f(x) = 2x^2 - 3x[/latex] on [latex][x, x + h][/latex]

For the following exercise, consider the graph of [latex]f[/latex] shown in the figure below.

5. Estimate the average rate of change from [latex]x = 2[/latex] to [latex]x = 5[/latex].

For the following exercises, use the graph of each function to estimate the intervals on which the function is increasing or decreasing.

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7. 

For the following exercises, find the average rate of change of each function on the interval specified.

8. [latex]h(x) = 5 - 2x^2[/latex] on [latex][-2, 4][/latex]
9. [latex]g(x) = 3x^3 - 1[/latex] on [latex][-3, 3][/latex]
10. [latex]p(t) = \dfrac{(t^2 - 4)(t + 1)}{t^2 + 3}[/latex] on [latex][-3, 1][/latex]

Real-World Applications.

11. A driver of a car stopped at a gas station to fill up their gas tank. They looked at their watch, and the time read exactly 3:40 p.m. At this time, they started pumping gas into the tank. At exactly 3:44, the tank was full and the driver noticed that they had pumped [latex]10.7[/latex] gallons. What is the average rate of flow of the gasoline into the gas tank?
12. The graph below illustrates the decay of a radioactive substance over [latex]t[/latex] days.
    
    Use the graph to estimate the average decay rate from [latex]t = 5[/latex] to [latex]t = 15[/latex].