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Chapter 1: Problem 31

Verify that \(f\) and \(g\) are inverse functions (a) algebraically and (b) graphically. $$f(x)=\frac{x-1}{x+5}, \quad g(x)=-\frac{5 x+1}{x-1}$$

Short Answer

Expert verified
If the \(f(g(x)) = x\) and \(g(f(x)) = x\), then these functions are inverse of each other algebraically. Graphically, if the plot of function \(f\) is a reflection of function \(g\) about the line \(y = x\), then these functions are inverse of each other.

Step by step solution

01

Algebraic Verification

First, we will perform the algebraic verification. This involves composing the function \(f\) into \(g\) and then \(g\) into \(f\), checking both times if the output equals to \(x\), which is the characteristic property of inverse functions. So we do the following: \n\n(i) Compute \(f(g(x))\). This means we replace everywhere in \(f(x)\) where there is an \(x\) with \(g(x)\). And then simplify.\n\n(ii) Compute \(g(f(x))\). This means we replace everywhere in \(g(x)\) where there is an \(x\) with \(f(x)\). And then simplify. If both compositions give \(x\), then the functions are indeed inverses of each other.
02

Graphical Verification

Next, we will perform the graphical verification. This requires plotting both functions on the same set of axis and check if the graph of \(f\) is a reflection of \(g\) about the line \(y = x\). If the functions are inverses, then one will be the reflection of the other about the line \(y = x\). This cannot be shown in this context, since this solution is text-based. However, you could use any graphing software to plot both functions along with the line \(y = x\) to verify that they are reflections of each other.

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