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Chapter 2: Problem 9

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=-x \text { and } g(x)=-x$$

Short Answer

Expert verified
The compositions of functions \(f(g(x))\) and \(g(f(x))\) both equal to \(x\), meaning that functions \(f(x)=-x\) and \(g(x)=-x\) are indeed inverses of each other.

Step by step solution

01

Understand the Functions

Here we have two functions, \(f(x)\) defined by \(f(x)=-x\) and \(g(x)\) defined by \(g(x)=-x\) .
02

Calculate \(f(g(x))\)

By substituting \(g(x)\) in function \(f\), we get \(f(g(x))=-(g(x))=-(-x)=x\)
03

Calculate \(g(f(x))\)

By substituting \(f(x)\) in function \(g\), we get\(g(f(x))=-(f(x))=-(-x)=x\)
04

Check If Functions are Inverses

If \(f(g(x)) = x\) and \(g(f(x)) = x\) for all \(x\) in the domain, then \(f\) and \(g\) are inverses of each other. Here, we have found that \(f(g(x)) = x\) and \(g(f(x)) = x\), meaning that \(f(x)\) and \(g(x)\) are indeed inverses of each other.

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