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Chapter 2: Problem 4

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=4 x+9 \text { and } g(x)=\frac{x-9}{4}$$

Short Answer

Expert verified
The functions \(f(g(x))=x\) and \(g(f(x))=x\) correctly demonstrate that the given functions \(f(x) = 4x+9\) and \(g(x) = \frac{x-9}{4}\) are inverses of each other.

Step by step solution

01

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

Substitute \(g(x)\) into function \(f(x)\). That is, wherever we see an \(x\) in \(f(x)\), replace it with \(g(x)\). \(f(g(x))=f\left(\frac{x-9}{4}\right) = 4 \cdot \frac{x-9}{4} + 9 = x\).
02

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

Substitute \(f(x)\) into function \(g(x)\). That is, wherever we see an \(x\) in \(g(x)\), replace it with \(f(x)\). \(g(f(x))=g(4x+9) = \frac{4x+9-9}{4} = x\).
03

Compare results

As both \(f(g(x))\) and \(g(f(x))\) are equal to \(x\), it is verified that \(f(x)\) and \(g(x)\) are inverses of each other.

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