Section 1.2 : Inverse Functions
3. Find the inverse for \(R\left( x \right) = {x^3} + 6\). 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 \(R\left( x \right)\), interchange the \(x\)’s and \(y\)’s, solve for \(y\) and the finally replace the \(y\) with \({R^{ - 1}}\left( x \right)\).
\[y = {x^3} + 6\]
Show Step 2
\[x = {y^3} + 6\]
Show Step 3
\[\begin{align*}x - 6 & = {y^3}\\ y &= \sqrt[3]{{x - 6}}\hspace{0.25in}\hspace{0.25in}\,\,\,\, \to \hspace{0.25in}\hspace{0.25in}\require{bbox} \bbox[2pt,border:1px solid black]{{{R^{ - 1}}\left( x \right) = \sqrt[3]{{x - 6}}}}\end{align*}\]
Finally, compute either \(\left( {R \circ {R^{ - 1}}} \right)\left( x \right)\) or \(\left( {{R^{ - 1}} \circ R} \right)\left( x \right)\) to verify our work.
Show Step 4Either composition can be done so let’s do \(\left( {{R^{ - 1}} \circ R} \right)\left( x \right)\) in this case.
\[\begin{align*}\left( {{R^{ - 1}} \circ R} \right)\left( x \right) & = {R^{ - 1}}\left[ {R\left( x \right)} \right]\\ & = \sqrt[3]{{\left( {{x^3} + 6} \right) - 6}}\\ & = \sqrt[3]{{{x^3}}}\\ & = x\end{align*}\]So, we got \(x\) out of the composition and so we know we’ve done our work correctly.