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Chapter 2: Problem 3

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=3 x+8 \text { and } g(x)=\frac{x-8}{3}$$

Short Answer

Expert verified
The function \(f(x) = 3x + 8\) and \(g(x) = \frac{x-8}{3}\) are inverses of each other as, for both \(f(g(x))\) and \(g(f(x))\), the result is \(x\).

Step by step solution

01

Calculate \(f(g(x))\)

First, we input \(g(x)\) into function \(f(x)\). Hence, \(f(g(x)) = f\left(\frac{x-8}{3}\right) = 3\left(\frac{x-8}{3}\right) + 8 = x + 8 - 8 = x.
02

Calculate \(g(f(x))\)

Secondly, input \(f(x)\) into \(g(x)\). Thus, \(g(f(x)) = g(3x+8) = \frac{3x + 8 - 8}{3} = x.
03

Confirm Inverse Relationship

Both \(f(g(x))\) and \(g(f(x))\) have resulted in \(x\), meaning these two functions are indeed inverses of each other.

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