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Chapter 1: 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
Yes, the functions \(f(x) = 6x\) and \(g(x) = \frac{x}{6}\) are inverses of each other since both \(f(g(x))\) and \(g(f(x))\) equals \(x\).

Step by step solution

01

Find \(f(g(x))\)

First, substitute \(g(x)\) into \(f(x)\). Because \(g(x) = \frac{x}{6}\), \(f(g(x))\) becomes \(f\left(\frac{x}{6}\right)\), which equals \(6\left(\frac{x}{6}\right)\). Then simplify that to get \(x\).
02

Find \(g(f(x))\)

Next, substitute \(f(x)\) into \(g(x)\). Since \(f(x) = 6x\), \(g(f(x))\) will be \(g(6x)\), which is equal to \(\frac{6x}{6}\). After simplifying, you get \(x\).
03

Check if \(f\) and \(g\) are inverses of each other

Both \(f(g(x))\) and \(g(f(x))\) equals \(x\), hence \(f\) and \(g\) are inverses of each other.

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