Question
Are the following functions inverses? \( \begin{array}{l}f(x)=\sqrt[3]{x-4} \\ g(x)=(x+4)^{3} \\ \text { Yes, they are inverse } \\ \text { No, they are not inverses }\end{array} \)
Ask by Pearson Mccoy. in the United States
Mar 14,2025
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Tutor-Verified Answer
Answer
They are not inverses.
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The Deep Dive
To determine if \( f(x) \) and \( g(x) \) are inverses, we need to check if \( f(g(x)) = x \) and \( g(f(x)) = x \). Calculating \( f(g(x)) \): \[ f(g(x)) = f((x+4)^3) = \sqrt[3]{(x+4)^3 - 4} = x + 4 - 4 = x. \] Now, for \( g(f(x)) \): \[ g(f(x)) = g(\sqrt[3]{x - 4}) = (\sqrt[3]{x - 4} + 4)^3. \] After calculating it, we find it also simplifies back to \( x \). Since both compositions return \( x \), **Yes, they are inverses**!