Section 1.2 : Inverse Functions
4. Find the inverse for \(g\left( x \right) = 4{\left( {x - 3} \right)^5} + 21\). Verify your inverse by computing one or both of the composition as discussed in this section.
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Hint : Remember the process described in this section. Replace the \(g\left( x \right)\), interchange the \(x\)’s and \(y\)’s, solve for \(y\) and the finally replace the \(y\) with \({g^{ - 1}}\left( x \right)\).
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\[y = 4{\left( {x - 3} \right)^5} + 21\]
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\[x = 4{\left( {y - 3} \right)^5} + 21\]
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\[\begin{align*}x - 21 & = 4{\left( {y - 3} \right)^5}\\ \frac{1}{4}\left( {x - 21} \right) & = {\left( {y - 3} \right)^5}\\ \sqrt[5]{{\frac{1}{4}\left( {x - 21} \right)}} & = y - 3\\ & y = 3 + \sqrt[5]{{\frac{1}{4}\left( {x - 21} \right)}}\hspace{0.25in}\hspace{0.25in}\,\,\,\, \to \hspace{0.25in}\hspace{0.25in}\require{bbox} \bbox[2pt,border:1px solid black]{{{g^{ - 1}}\left( x \right) = 3 + \sqrt[5]{{\frac{1}{4}\left( {x - 21} \right)}}}}\end{align*}\]
Finally, compute either \(\left( {g \circ {g^{ - 1}}} \right)\left( x \right)\) or \(\left( {{g^{ - 1}} \circ g} \right)\left( x \right)\) to verify our work.
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Either composition can be done so let’s do \(\left( {g \circ {g^{ - 1}}} \right)\left( x \right)\) in this case.
\[\begin{align*}\left( {g \circ {g^{ - 1}}} \right)\left( x \right) & = g\left[ {{g^{ - 1}}\left( x \right)} \right]\\ & = 4{\left( {\left[ {3 + \sqrt[5]{{\frac{1}{4}\left( {x - 21} \right)}}} \right] - 3} \right)^5} + 21\\ & = 4{\left( {\sqrt[5]{{\frac{1}{4}\left( {x - 21} \right)}}} \right)^5} + 21\\ & = 4\left( {\frac{1}{4}\left( {x - 21} \right)} \right) + 21\\ & = \left( {x - 21} \right) + 21\\ & = x\end{align*}\]So, we got \(x\) out of the composition and so we know we’ve done our work correctly.