Stewart Precalc 6e Section 2.7: Combining Functions

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Stewart Precalc 6e Section 2.7: Combining Functions 0/79
1 Concept - Definition of one-to-one · Level 1
A function \(f\) is one-to-one if different inputs produce ______ outputs. You can tell from the graph that a function is one-to-one by using the ______ Test.
2 Concept - Existence of an inverse · Level 1
(a) For a function to have an inverse, it must be ______. So which one of the following functions has an inverse: \(f(x) = x^2\) or \(g(x) = x^3\)? (b) What is the inverse of the function that you chose in part (a)?
3 Concept - Verbal and algebraic description · Level 2
A function \(f\) has the following verbal description: 'Multiply by 3, add 5, and then take the third power of the result.' (a) Write a verbal description for \(f^{-1}\). (b) Find algebraic formulas that express \(f\) and \(f^{-1}\) in terms of the input \(x\).
4 Concept - True/False properties · Level 1
True or false? (a) If \(f\) has an inverse, then \(f^{-1}(x)\) is the same as \(\dfrac{1}{f(x)}\). (b) If \(f\) has an inverse, then \(f^{-1}(f(x)) = x\).
5 Determine one-to-one from a graph · Level 1
The graph of a function \(f\) is given. Determine whether \(f\) is one-to-one.
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6 Determine one-to-one from a graph · Level 1
The graph of a function \(f\) is given. Determine whether \(f\) is one-to-one.
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7 Determine one-to-one from a graph · Level 1
The graph of a function \(f\) is given. Determine whether \(f\) is one-to-one.
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8 Determine one-to-one from a graph · Level 1
The graph of a function \(f\) is given. Determine whether \(f\) is one-to-one.
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9 Determine one-to-one from a graph · Level 1
The graph of a function \(f\) is given. Determine whether \(f\) is one-to-one.
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10 Determine one-to-one from a graph · Level 1
The graph of a function \(f\) is given. Determine whether \(f\) is one-to-one.
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11 Determine one-to-one algebraically · Level 1
Determine whether the function is one-to-one: \(f(x) = -2x + 4\).
12 Determine one-to-one algebraically · Level 1
Determine whether the function is one-to-one: \(f(x) = 3x - 2\).
13 Determine one-to-one algebraically · Level 1
Determine whether the function is one-to-one: \(g(x) = \sqrt{x}\).
14 Determine one-to-one algebraically · Level 1
Determine whether the function is one-to-one: \(q(x) = |x|\).
15 Determine one-to-one algebraically · Level 1
Determine whether the function is one-to-one: \(h(x) = x^2 - 2x\).
16 Determine one-to-one algebraically · Level 1
Determine whether the function is one-to-one: \(h(x) = x^3 + 8\).
17 Determine one-to-one algebraically · Level 1
Determine whether the function is one-to-one: \(f(x) = x^4 + 5\).
18 Determine one-to-one algebraically · Level 2
Determine whether the function is one-to-one: \(f(x) = x^4 + 5\), \(0 \leq x \leq 2\).
19 Determine one-to-one algebraically · Level 1
Determine whether the function is one-to-one: \(f(x) = \dfrac{1}{x^2}\).
20 Determine one-to-one algebraically · Level 1
Determine whether the function is one-to-one: \(f(x) = \dfrac{1}{x}\).
21 Inverse function values · Level 1
Assume that \(f\) is a one-to-one function. (a) If \(f(2) = 7\), find \(f^{-1}(7)\). (b) If \(f^{-1}(3) = -1\), find \(f(-1)\).
22 Inverse function values · Level 1
Assume that \(f\) is a one-to-one function. (a) If \(f(5) = 18\), find \(f^{-1}(18)\). (b) If \(f^{-1}(4) = 2\), find \(f(2)\).
23 Find a specific inverse value · Level 2
If \(f(x) = 5 - 2x\), find \(f^{-1}(3)\).
24 Find a specific inverse value · Level 2
If \(g(x) = x^2 + 4x\) with \(x \geq -2\), find \(g^{-1}(5)\).
25 Verify inverse functions · Level 2
Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other: \(f(x) = x - 6\); \(g(x) = x + 6\).
26 Verify inverse functions · Level 2
Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other: \(f(x) = 3x\); \(g(x) = \dfrac{x}{3}\).
27 Verify inverse functions · Level 2
Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other: \(f(x) = 2x - 5\); \(g(x) = \dfrac{x + 5}{2}\).
28 Verify inverse functions · Level 2
Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other: \(f(x) = \dfrac{3 - x}{4}\); \(g(x) = 3 - 4x\).
29 Verify inverse functions · Level 2
Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other: \(f(x) = \dfrac{1}{x}\); \(g(x) = \dfrac{1}{x}\).
30 Verify inverse functions · Level 2
Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other: \(f(x) = x^5\); \(g(x) = \sqrt[5]{x}\).
31 Verify inverse functions · Level 2
Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other: \(f(x) = x^2 - 4\), \(x \geq 0\); \(g(x) = \sqrt{x + 4}\), \(x \geq -4\).
32 Verify inverse functions · Level 2
Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other: \(f(x) = x^3 + 1\); \(g(x) = (x - 1)^{\dfrac{1}{3}}\).
33 Verify inverse functions · Level 2
Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other: \(f(x) = \dfrac{1}{x - 1}\), \(x \neq 1\); \(g(x) = \dfrac{1}{x} + 1\), \(x \neq 0\).
34 Verify inverse functions · Level 2
Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other: \(f(x) = \sqrt{4 - x^2}\), \(0 \leq x \leq 2\); \(g(x) = \sqrt{4 - x^2}\), \(0 \leq x \leq 2\).
35 Verify inverse functions · Level 3
Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other: \(f(x) = \dfrac{x + 2}{x - 2}\); \(g(x) = \dfrac{2x + 2}{x - 1}\).
36 Verify inverse functions · Level 3
Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other: \(f(x) = \dfrac{x - 5}{3x + 4}\); \(g(x) = \dfrac{5 + 4x}{1 - 3x}\).
37 Find inverse function · Level 1
Find the inverse function of \(f\): \(f(x) = 2x + 1\).
38 Find inverse function · Level 1
Find the inverse function of \(f\): \(f(x) = 6 - x\).
39 Find inverse function · Level 1
Find the inverse function of \(f\): \(f(x) = 4x + 7\).
40 Find inverse function · Level 1
Find the inverse function of \(f\): \(f(x) = 3 - 5x\).
41 Find inverse function · Level 2
Find the inverse function of \(f\): \(f(x) = 5 - 4x^3\).
42 Find inverse function · Level 2
Find the inverse function of \(f\): \(f(x) = \dfrac{1}{x^2}\), \(x > 0\).
43 Find inverse function · Level 2
Find the inverse function of \(f\): \(f(x) = \dfrac{1}{x + 2}\).
44 Find inverse function · Level 2
Find the inverse function of \(f\): \(f(x) = \dfrac{x - 2}{x + 2}\).
45 Find inverse function · Level 2
Find the inverse function of \(f\): \(f(x) = \dfrac{x}{x + 4}\).
46 Find inverse function · Level 2
Find the inverse function of \(f\): \(f(x) = \dfrac{3x}{x - 2}\).
47 Find inverse function · Level 2
Find the inverse function of \(f\): \(f(x) = \dfrac{2x + 5}{x - 7}\).
48 Find inverse function · Level 2
Find the inverse function of \(f\): \(f(x) = \dfrac{4x - 2}{3x + 1}\).
49 Find inverse function · Level 2
Find the inverse function of \(f\): \(f(x) = \dfrac{1 + 3x}{5 - 2x}\).
50 Find inverse function · Level 2
Find the inverse function of \(f\): \(f(x) = \dfrac{2x - 1}{x - 3}\).
51 Find inverse function · Level 2
Find the inverse function of \(f\): \(f(x) = \sqrt{2 + 5x}\).
52 Find inverse function · Level 3
Find the inverse function of \(f\): \(f(x) = x^2 + x\), \(x \geq -\dfrac{1}{2}\).
53 Find inverse function · Level 2
Find the inverse function of \(f\): \(f(x) = 4 - x^2\), \(x \geq 0\).
54 Find inverse function · Level 2
Find the inverse function of \(f\): \(f(x) = \sqrt{2x - 1}\).
55 Find inverse function · Level 2
Find the inverse function of \(f\): \(f(x) = 4 + \sqrt[3]{x}\).
56 Find inverse function · Level 3
Find the inverse function of \(f\): \(f(x) = (2 - x^3)^5\).
57 Find inverse function · Level 2
Find the inverse function of \(f\): \(f(x) = 1 + \sqrt{1 + x}\).
58 Find inverse function · Level 2
Find the inverse function of \(f\): \(f(x) = \sqrt{9 - x^2}\), \(0 \leq x \leq 3\).
59 Find inverse function · Level 1
Find the inverse function of \(f\): \(f(x) = x^4\), \(x \geq 0\).
60 Find inverse function · Level 1
Find the inverse function of \(f\): \(f(x) = 1 - x^3\).
61 Graph and find inverse · Level 2
A function \(f\) is given. (a) Sketch the graph of \(f\). (b) Use the graph of \(f\) to sketch the graph of \(f^{-1}\). (c) Find \(f^{-1}\). \(f(x) = 3x - 6\).
62 Graph and find inverse · Level 2
A function \(f\) is given. (a) Sketch the graph of \(f\). (b) Use the graph of \(f\) to sketch the graph of \(f^{-1}\). (c) Find \(f^{-1}\). \(f(x) = 16 - x^2\), \(x \geq 0\).
63 Graph and find inverse · Level 2
A function \(f\) is given. (a) Sketch the graph of \(f\). (b) Use the graph of \(f\) to sketch the graph of \(f^{-1}\). (c) Find \(f^{-1}\). \(f(x) = \sqrt{x + 1}\).
64 Graph and find inverse · Level 2
A function \(f\) is given. (a) Sketch the graph of \(f\). (b) Use the graph of \(f\) to sketch the graph of \(f^{-1}\). (c) Find \(f^{-1}\). \(f(x) = x^3 - 1\).
65 Determine one-to-one using graph · Level 2
Draw the graph of \(f\) and use it to determine whether the function is one-to-one: \(f(x) = x^3 - x\).
66 Determine one-to-one using graph · Level 2
Draw the graph of \(f\) and use it to determine whether the function is one-to-one: \(f(x) = x^3 + x\).
67 Determine one-to-one using graph · Level 2
Draw the graph of \(f\) and use it to determine whether the function is one-to-one: \(f(x) = \dfrac{x + 12}{x - 6}\).
68 Determine one-to-one using graph · Level 3
Draw the graph of \(f\) and use it to determine whether the function is one-to-one: \(f(x) = \sqrt{x^3 - 4x + 1}\).
69 Determine one-to-one using graph · Level 3
Draw the graph of \(f\) and use it to determine whether the function is one-to-one: \(f(x) = |x| - |x - 6|\).
70 Determine one-to-one using graph · Level 3
Draw the graph of \(f\) and use it to determine whether the function is one-to-one: \(f(x) = x \cdot |x|\).
71 Example - Combinations of Functions and Their Domains · Level 2
Let \(f(x) = \dfrac{1}{x - 2}\) and \(g(x) = \sqrt{x}\).
(a) Find the functions \(f + g\), \(f - g\), \(f g\), and \(\dfrac{f}{g}\) and their domains.
(b) Find \((f + g)(4)\), \((f - g)(4)\), \((f g)(4)\), and \(\left(\dfrac{f}{g}\right)(4)\).

Enter your answer directly below each part above.

72 Example - Using Graphical Addition · Level 2
The graphs of \(f\) and \(g\) are shown in Figure 1. Use graphical addition to graph the function \(f + g\).
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73 Example - Finding the Composition of Functions · Level 2
Let \(f(x) = x^2\) and \(g(x) = x - 3\).
(a) Find the functions \(f circle.small g\) and \(g circle.small f\) and their domains.
(b) Find \((f circle.small g)(5)\) and \((g circle.small f)(7)\).

Enter your answer directly below each part above.

74 Example - Finding the Composition of Functions · Level 3
If \(f(x) = \sqrt{x}\) and \(g(x) = \sqrt{2 - x}\), find the following functions and their domains.
(a) \(f circle.small g\)
(b) \(g circle.small f\)
(c) \(f circle.small f\)
(d) \(g circle.small g\)

Enter your answer directly below each part above.

75 Example - A Composition of Three Functions · Level 3
Find \(f circle.small g circle.small h\) if \(f(x) = \dfrac{x}{x + 1}\), \(g(x) = x^{10}\), and \(h(x) = x + 3\).
76 Example - Recognizing a Composition of Functions · Level 2
Given \(F(x) = \sqrt[4]{x + 9}\), find functions \(f\) and \(g\) such that \(F = f circle.small g\).
77 Example - An Application of Composition of Functions · Level 3
A ship is traveling at 20 mi/h parallel to a straight shoreline. The ship is 5 mi from shore. It passes a lighthouse at noon.
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(a) Express the distance \(s\) between the lighthouse and the ship as a function of \(d\), the distance the ship has traveled since noon; that is, find \(f\) so that \(s = f(d)\).
(b) Express \(d\) as a function of \(t\), the time elapsed since noon; that is, find \(g\) so that \(d = g(t)\).
(c) Find \(f circle.small g\). What does this function represent?

Enter your answer directly below each part above.

78 Example - Finding inverse of a rational function · Level 2
Find the inverse of the function \(f(x) = \dfrac{2x + 3}{x - 1}\).
79 Example - Graphing the inverse of a function · Level 3
(a) Sketch the graph of \(f(x) = \sqrt{x - 2}\). (b) Use the graph of \(f\) to sketch the graph of \(f^{-1}\). (c) Find an equation for \(f^{-1}\).
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