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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 composed functions \(f(g(x))\) and \(g(f(x))\) both yield \(x\), thus demonstrating that \(f\) and \(g\) are indeed inverses of each other.

Step by step solution

01

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

To compute \(f(g(x))\), replace the \(x\) in function \(f\) with function \(g(x)\). This gives us \(f(g(x)) = f\left(\frac{x}{6}\right)\). Now, substitute \(f(x)\) in the equation with the given \(f(x)=6x\). So, \(f(g(x)) = 6 \left(\frac{x}{6}\right)\). After multiplication, we get \(f(g(x))=x\).
02

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

Proceeding similarly, to compute \(g(f(x))\), replace \(x\) in function \(g\) with function \(f(x)\). This gives \(g(f(x)) = g(6x)\). Substituting \(g(x)\) with the given \(g(x)=\frac{x}{6}\), gives us \(g(f(x)) = \frac{6x}{6}\). After simplification, this too gives \(g(f(x))=x\).
03

Determine if \(f\) and \(g\) are inverses

Since both \(f(g(x))\) and \(g(f(x))\) resulted in \(x\), we can conclude that \(f\) and \(g\) are indeed inverses of each other. Two functions are inverses of each other if the composition of both functions in any order results in \(x\).

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Most popular questions from this chapter

use a graphing utility to graph each function. Use \(a[-5,5,1]\) by \([-5,5,1]\) viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant. $$ g(x)=\left|4-x^{2}\right| $$

Sketch the graph of \(f\) using the following properties. (More than one correct graph is possible.) \(f\) is a piecewise function that is decreasing on \((-\infty, 2), f(2)=0, f\) is increasing on \((2, \infty),\) and the range of \(f\) is \([0, \infty)\)

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