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Chapter 1: Problem 26

Verify that \(f\) and \(g\) are inverse functions (a) algebraically and (b) graphically. $$f(x)=\frac{1}{x}, \quad g(x)=\frac{1}{x}$$

Short Answer

Expert verified
The functions \(f(x)\) and \(g(x)\) are inverses of each other. This has been verified both algebraically, as both \(f(g(x))\) and \(g(f(x))\) resulted in the identity function \(x\), and graphically, as the graphs of \(f(x)\) and \(g(x)\) are reflections over the line \(y=x\).

Step by step solution

01

Algebraic Verification

First, verify algebraically by finding the compositions \(f(g(x))\) and \(g(f(x))\). Since \(f(x)\) and \(g(x)\) are both defined as \(\frac{1}{x}\), we have: \[f(g(x))=f\left(\frac{1}{x}\right)=\frac{1}{\frac{1}{x}}=x,\] Similarly, \[g(f(x))=g\left(\frac{1}{x}\right)=\frac{1}{\frac{1}{x}}=x.\] Thus, both \(f(g(x))\) and \(g(f(x))\) evaluate to \(x\), so \(f\) and \(g\) are inverse functions algebraically.
02

Graphical Verification

The graphs of both functions are hyperbolas reflected over the line \(y=x\). Since the two hyperbolas are the same reflection, the two functions are inverses of each other graphically as well.

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