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Chapter 1: Problem 32

Verify that \(f\) and \(g\) are inverse functions (a) algebraically and (b) graphically. $$f(x)=\frac{x+3}{x-2}, \quad g(x)=\frac{2 x+3}{x-1}$$

Short Answer

Expert verified
The functions \(f(x)\) and \(g(x)\) are verified to be inverse functions both algebraically and graphically.

Step by step solution

01

Algebraic Verification

To algebraically verify that the functions are inverses, compute \(f(g(x))\) and \(g(f(x))\) and check whether both equal \(x\). Start with \(f(g(x))=\frac{(2x+3)/(x-1)+3}{(2x+3)/(x-1)-2}\). This simplifies to \(x\). Then compute \(g(f(x))=\frac{2((x+3)/(x-2))+3}{(x+3)/(x-2)-1}\). This also simplifies to \(x\). Thus, \(f\) and \(g\) are inverse functions algebraically.
02

Graphical Verification

To graphically verify that the functions are inverses, plot the graphs of \(f(x)\), \(g(x)\), and the line \(y = x\) on the same axes. Note that \(f(x)\) and \(g(x)\) will be reflections of each other across the line \(y = x\), indicating that they are indeed inverse functions.

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