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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 \(f(g(x))\) and \(g(f(x))\) both equal to \(x\), so functions \(f(x)=-x\) and \(g(x)=-x\) are inverses of each other.

Step by step solution

01

Find the Composition f(g(x))

We substitute \(g(x)\) into the function \(f(x)\). Since \(f(x)=-x\) and \(g(x)=-x\), we have \(f(g(x))=-(-x)\). When the negative of \(-x\) is taken, it becomes \(x\). So, \(f(g(x))=x\).
02

Find the Composition g(f(x))

We substitute \(f(x)\) into the function \(g(x)\). Since \(f(x)=-x\) and \(g(x)=-x\), we have \(g(f(x))=-(-x)\). The negative of \(-x\) is (\(x\)), so \(g(f(x))=x\).
03

Check if f and g are Inverses

Both \(f(g(x))\) and \(g(f(x))\) are equal to \(x\). Therefore, the given functions \(f(x)=-x\) and \(g(x)=-x\) are inverses of each other.

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