For each pair of functions f and g below, find f(g (x)) and g(f(x)). Then, determine whether fand g are inverses of each other. Simplify your answers as much as possible. (Assume that your expressions are defined for all x in the domain of the composition. You do not have to indicate the domain.) (a) f(*) = -, x + 0 (b) f(x) = x + 6 g (x) = g (x) = -x + 6 %3D se(4)) = | re (+)) = D g(+)) = 0 O f and g are inverses of each other Of and g are inverses of each other Of and g are not inverses of each other O f and g are not inverses of each other

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Question
100%
### Composition of Functions and Inverses

#### Instructions
For each pair of functions \( f \) and \( g \) below:
1. Find \( f(g(x)) \) and \( g(f(x)) \).
2. Determine whether \( f \) and \( g \) are inverses of each other.
3. Simplify your answers as much as possible. (Assume that your expressions are defined for all \( x \) in the domain of the composition. You do not have to indicate the domain.)

---

#### (a)
- \( f(x) = \frac{4}{x} , \, x \neq 0 \)
- \( g(x) = \frac{4}{x} , \, x \neq 0 \)

Calculate:
- \( f(g(x)) = \) (box for answer)
- \( g(f(x)) = \) (box for answer)

Determine:
- \( f \) and \( g \) are inverses of each other. (Selectable option)
- \( f \) and \( g \) are not inverses of each other. (Selectable option)

---

#### (b)
- \( f(x) = x + 6 \)
- \( g(x) = x - 6 \)

Calculate:
- \( f(g(x)) = \) (box for answer)
- \( g(f(x)) = \) (box for answer)

Determine:
- \( f \) and \( g \) are inverses of each other. (Selectable option)
- \( f \) and \( g \) are not inverses of each other. (Selectable option)

---

#### Additional Tools

In the upper right corner, there are tools for:
- Resetting the problem.
- Reviewing hints.
- Accessing additional help.

#### Submission
- Press "Check" to verify your answers.

> © 2021 McGraw Hill LLC. All Rights Reserved.
Transcribed Image Text:### Composition of Functions and Inverses #### Instructions For each pair of functions \( f \) and \( g \) below: 1. Find \( f(g(x)) \) and \( g(f(x)) \). 2. Determine whether \( f \) and \( g \) are inverses of each other. 3. Simplify your answers as much as possible. (Assume that your expressions are defined for all \( x \) in the domain of the composition. You do not have to indicate the domain.) --- #### (a) - \( f(x) = \frac{4}{x} , \, x \neq 0 \) - \( g(x) = \frac{4}{x} , \, x \neq 0 \) Calculate: - \( f(g(x)) = \) (box for answer) - \( g(f(x)) = \) (box for answer) Determine: - \( f \) and \( g \) are inverses of each other. (Selectable option) - \( f \) and \( g \) are not inverses of each other. (Selectable option) --- #### (b) - \( f(x) = x + 6 \) - \( g(x) = x - 6 \) Calculate: - \( f(g(x)) = \) (box for answer) - \( g(f(x)) = \) (box for answer) Determine: - \( f \) and \( g \) are inverses of each other. (Selectable option) - \( f \) and \( g \) are not inverses of each other. (Selectable option) --- #### Additional Tools In the upper right corner, there are tools for: - Resetting the problem. - Reviewing hints. - Accessing additional help. #### Submission - Press "Check" to verify your answers. > © 2021 McGraw Hill LLC. All Rights Reserved.
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