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Chapter 1: Problem 28

Verify that \(f\) and \(g\) are inverse functions (a) algebraically and (b) graphically. $$f(x)=1-x^{3}, \quad g(x)=\sqrt[3]{1-x}$$

Short Answer

Expert verified
After calculations, we have \(f(g(x)) = x\) and \(g(f(x)) = x\) which satisfies the criteria of inverse functions. Also, on drawing the function graphs, they are mirror images of each other about the line \(y = x\). Therefore, \(f\) and \(g\) are inverses of each other.

Step by step solution

01

Verify Algebraically

Firstly, we will verify that \(f(g(x)) = x\) and \(g(f(x)) = x\). This is the criteria for determining if two functions are inverses of each other algebraically. If both \(f(g(x))\) and \(g(f(x))\) equal \(x\), it means that \(f\) and \(g\) are inverse functions of each other. Let's calculate \(f(g(x))\) and \(g(f(x))\).
02

Calculate f(g(x)) and g(f(x))

Plug \(g(x)\) into \(f(x)\) to get \(f(g(x))\) and likewise plug \(f(x)\) into \(g(x)\) to get \(g(f(x))\). \[f(g(x)) = f(\sqrt[3]{1-x}) = 1-(\sqrt[3]{1-x})^{3} = 1-(1-x) = x\] Therefore, \(f(g(x)) = x.\) Similarly, calculate \(g(f(x))\), \[g(f(x)) = g(1-x^{3}) = \sqrt[3]{1-(1-x^{3})} = \sqrt[3]{x^{3}} = x.\] We have proven that \(f(g(x)) = x\) and \(g(f(x)) = x\) algebraically, therefore \(f\) and \(g\) are inverse functions of each other.
03

Verify Graphically

Graphically two functions are inverses if and only if their graphs are mirror images of each other about the line \(y = x\). Draw \(y = f(x)\), \(y = g(x)\), and the line \(y = x\). Notice if the graphs of \(f\) and \(g\) are mirror images of each other about the line \(y = x\), hence confirming the inverse relationship.

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