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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
Yes, the functions \(f(x) = 4x + 9\) and \(g(x) = (x - 9)/4\) are inverses of each other as \(f(g(x)) = x\) and \(g(f(x)) = x\).

Step by step solution

01

Compute \(f(g(x))\)

To compute \(f(g(x))\), substitute \(g(x)\) into \(f(x)\). Therefore, \(f(g(x)) = f\left( \frac{x-9}{4}\right)=4\left(\frac{x-9}{4}\right)+9 =x.\) Thus, \(f(g(x))=x\).
02

Compute \(g(f(x))\)

To compute \(g(f(x))\), substitute \(f(x)\) into \(g(x)\). Therefore, \(g(f(x)) = g(4x+9) = \frac{4x+9-9}{4}=x.\) Thus, \(g(f(x))=x\).
03

Determine if \(f\) and \(g\) are inverses

For functions \(f\) and \(g\) to be inverses of each other, both composites \(f(g(x))\) and \(g(f(x))\) must be equal to \(x\). Here, we have both \(f(g(x))=x\) and \(g(f(x))=x\). Thus, the functions \(f\) and \(g\) are indeed inverses of each other.

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