For each pair of functions f and g below, find f(g (x)) and g (f(x)). Then, determine whether f and 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) = - - (b) f(x) = 2x + 7 %3D %3D 4 7 X. g (x) = -4x g(x) = 2 sl8(x)) = | sl8 (x)) = [ %3D %3D 8( («)) = ] 8( (x)) = [ %3D %3D Of and g are inverses of each other Of and g are inverses of each other Of and g are not inverses of each other Of and g are not inverses of each other
For each pair of functions f and g below, find f(g (x)) and g (f(x)). Then, determine whether f and 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) = - - (b) f(x) = 2x + 7 %3D %3D 4 7 X. g (x) = -4x g(x) = 2 sl8(x)) = | sl8 (x)) = [ %3D %3D 8( («)) = ] 8( (x)) = [ %3D %3D Of and g are inverses of each other Of and g are inverses of each other Of and g are not inverses of each other Of 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
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![**Determining Whether Two Functions are Inverses**
For each pair of functions \( f \) and \( g \) below, find \( f(g(x)) \) and \( g(f(x)) \).
Then, determine whether \( f \) and \( g \) 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)**
\( f(x) = -\frac{x}{4} \)
\( g(x) = -4x \)
- \( f(g(x)) = \) [ ]
- \( g(f(x)) = \) [ ]
[ ] \( f \) and \( g \) are inverses of each other
[ ] \( f \) and \( g \) are not inverses of each other
**(b)**
\( f(x) = 2x + 7 \)
\( g(x) = \frac{x - 7}{2} \)
- \( f(g(x)) = \) [ ]
- \( g(f(x)) = \) [ ]
[ ] \( f \) and \( g \) are inverses of each other
[ ] \( f \) and \( g \) are not inverses of each other
**Instructions**: Use the spaces provided to calculate the compositions \( f(g(x)) \) and \( g(f(x)) \). After simplifying the expressions, check whether each composition equals \( x \). If both compositions equal \( x \), then \( f \) and \( g \) are inverses. If not, they are not inverses.
**Interactive Elements**: Click "Explanation" for a walkthrough of solving each part or "Check" to verify your answers.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F90b823f6-a59b-43f1-924e-a48b4e93719a%2F3d9ffeab-ce50-4c1f-bc0c-3ac2f8ae52ff%2Fhxoeb8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Determining Whether Two Functions are Inverses**
For each pair of functions \( f \) and \( g \) below, find \( f(g(x)) \) and \( g(f(x)) \).
Then, determine whether \( f \) and \( g \) 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)**
\( f(x) = -\frac{x}{4} \)
\( g(x) = -4x \)
- \( f(g(x)) = \) [ ]
- \( g(f(x)) = \) [ ]
[ ] \( f \) and \( g \) are inverses of each other
[ ] \( f \) and \( g \) are not inverses of each other
**(b)**
\( f(x) = 2x + 7 \)
\( g(x) = \frac{x - 7}{2} \)
- \( f(g(x)) = \) [ ]
- \( g(f(x)) = \) [ ]
[ ] \( f \) and \( g \) are inverses of each other
[ ] \( f \) and \( g \) are not inverses of each other
**Instructions**: Use the spaces provided to calculate the compositions \( f(g(x)) \) and \( g(f(x)) \). After simplifying the expressions, check whether each composition equals \( x \). If both compositions equal \( x \), then \( f \) and \( g \) are inverses. If not, they are not inverses.
**Interactive Elements**: Click "Explanation" for a walkthrough of solving each part or "Check" to verify your answers.
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