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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 computed function compositions are \(f(g(x)) = x\) and \(g(f(x)) = x\). Since they both match the identity function, it can be concluded that the functions \(f(x)\) and \(g(x)\) are inverses of each other.

Step by step solution

01

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

Let's substitute \(g(x)\) into \(f(x)\). So, \(f(g(x)) = f(\frac{x}{6}) = 6 * \frac{x}{6} = x\)
02

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

Now, let's do the opposite and substitute \(f(x)\) into \(g(x)\). So, \(g(f(x)) = g(6x) = \frac{6x}{6} = x\)
03

Compare the Compositions with the identity function

Both \(f(g(x))\) and \(g(f(x))\) resulted in \(x\), which is the same as the identity function \(I(x) = x\). Thus, functions \(f(x)\) and \(g(x)\) are inverses of each other.

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