Problem 6 - 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. $$f(x)=3 x-7 \text { and } g(x)=\frac{x+3}{7}$$

Step-by-Step Solution

Answer

The composite functions are: \(f(g(x))=\frac{3x-40}{7}\) and \(g(f(x))=\frac{3x-4}{7}\). And functions \(f(x)\) and \(g(x)\) are not inverses of each other.

Step 1: Finding \(f(g(x))\)

To find the composite function \(f(g(x))\), substitute the function \(g(x)\) into \(f(x)\). That is: \(f(g(x))=f(\frac{x+3}{7})=3(\frac{x+3}{7})-7=\frac{3x+9}{7}-7\). Furthermore, simplify this to get \(\frac{3x+9-49}{7}=\frac{3x-40}{7}\)

Step 2: Finding \(g(f(x))\)

To find the composite function \(g(f(x))\), substitute the function \(f(x)\) into \(g(x)\). That is: \(g(f(x))=g(3x-7)=\frac{3x-7+3}{7}=\frac{3x-4}{7}\). Now, this needs to be simplified. But it turns out there's nothing to simplify further.

Step 3: Determine Inverses

To find if \(f(x)\) and \(g(x)\) are inverses, \(f(g(x))\) and \(g(f(x))\) must return \(x\). But here, \(f(g(x))=\frac{3x-40}{7}\) and \(g(f(x))=\frac{3x-4}{7}\), which do not return \(x\). So, functions \(f(x)\) and \(g(x)\) are not inverses of each other.