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Chapter 1: Problem 6

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=3 x-7 \text { and } g(x)=\frac{x+3}{7}$$

Short Answer

Expert verified
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 by step solution

01

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 \(\frac{3x+9-49}{7}=\frac{3x-40}{7}\)
02

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.
03

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.

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