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Chapter 2: Problem 2

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=6 x \text { and } g(x)=\frac{x}{6}$$

Short Answer

Expert verified
The function compositions \(f(g(x)) = x\) and \(g(f(x)) = x\), so the functions \(f(x) = 6x\) and \(g(x) = \frac{x}{6}\) are inverses of each other.

Step by step solution

01

Compute \(f(g(x))\)

The function \(g(x) = \frac{x}{6}\) is being inserted into the function \(f(x)\). So, \(f(g(x)) = f\left(\frac{x}{6}\right)\). Substituting \(f(x) = 6x\) gives us \(f(g(x)) = 6 \cdot \frac{x}{6} = x\).
02

Compute \(g(f(x))\)

In this part, the function \(f(x) = 6x\) is being inserted into the function \(g(x)\). So, \(g(f(x)) = g(6x)\). Substituting \(g(x) = \frac{x}{6}\) gives us \(g(f(x)) = \frac{6x}{6} = x\).
03

Determine if the functions are inverses

Functions f and g are inverses of each other if the composition \(f(g(x))\) and \(g(f(x))\) both equal to x, which in this case, both do. Hence, f and g are inverses of each other.

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