Problem 4 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

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 1: 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: \[f(g(x)) = x\]

Step 2: 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: \[g(f(x)) = x\]

Step 3: 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 ‘x’. 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.