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Chapter 1: 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 \(f(x) = 4x + 9\) and \(g(x) = \frac{x-9}{4}\) are found to be inverses of each other, as both \(f(g(x))\) and \(g(f(x))\) are equal to 'x'.

Step by step solution

01

Finding \(f(g(x))\)

First, substitute function \(g(x)\) into function \(f(x)\): \[f(g(x))=f(\frac{x-9}{4})\] Now replace \(x\) in \(f(x)\) with \(\frac{x-9}{4}\): \[f(g(x))=4(\frac{x-9}{4})+9\] Simplify the equation: \[f(g(x)) = x\]
02

Finding \(g(f(x))\)

The next step involves doing the reverse: substitute function \(f(x)\) into function \(g(x)\). So, \[g(f(x))=g(4x+9)\] Now replace \(x\) in \(g(x)\) with \(4x+9\): \[g(f(x))=\frac{(4x+9)-9}{4}\] Simplify the equation: \[g(f(x)) = x\]
03

Determining inverses

The final task is to determine whether or not the two functions are inverses of each other. To bring this out, compare the results of step 1 and step 2. Both \(f(g(x))\) and \(g(f(x))\) equal 鈥榵鈥�. Therefore, it can be said that the functions \(f(x) = 4x + 9\) and \(g(x) = \frac{x-9}{4}\) are indeed inverses of each other.

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