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### Section 1.2 : Inverse Functions

6. Find the inverse for $$f\left( x \right) = \sqrt[7]{{5x + 8}}$$. 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 $$f\left( x \right)$$, interchange the $$x$$’s and $$y$$’s, solve for $$y$$ and the finally replace the $$y$$ with $${f^{ - 1}}\left( x \right)$$.
Start Solution
$y = \sqrt[7]{{5x + 8}}$ Show Step 2
$x = \sqrt[7]{{5y + 8}}$ Show Step 3
\begin{align*}x & = \sqrt[7]{{5y + 8}}\\ {x^7} & = 5y + 8\\ & {x^7} - 8 = 5y\\ y & = \frac{1}{5}\left( {{x^7} - 8} \right)\hspace{0.25in}\hspace{0.25in}\,\,\,\, \to \hspace{0.25in}\hspace{0.25in}\require{bbox} \bbox[2pt,border:1px solid black]{{{f^{ - 1}}\left( x \right) = \frac{1}{5}\left( {{x^7} - 8} \right)}}\end{align*}

Finally, compute either $$\left( {f \circ {f^{ - 1}}} \right)\left( x \right)$$ or $$\left( {{f^{ - 1}} \circ f} \right)\left( x \right)$$ to verify our work.

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Either composition can be done so let’s do $$\left( {f \circ {f^{ - 1}}} \right)\left( x \right)$$ in this case.

\begin{align*}\left( {f \circ {f^{ - 1}}} \right)\left( x \right) & = f\left[ {{f^{ - 1}}\left( x \right)} \right]\\ & = \sqrt[7]{{5\left[ {\frac{1}{5}\left( {{x^7} - 8} \right)} \right] + 8}}\\ & = \sqrt[7]{{\left[ {{x^7} - 8} \right] + 8}}\\ & = \sqrt[7]{{{x^7}}}\\ & = x\end{align*}

So, we got $$x$$ out of the composition and so we know we’ve done our work correctly.