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Section 3-7 : Inverse Functions

1. Given \(h\left( x \right) = 5 - 9x\) find \({h^{ - 1}}\left( x \right)\) .

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Hint : Just follow the process outlines in the notes and you’ll be set to do this problem!
Start Solution

For the first step we simply replace the function with a \(y\).

\[y = 5 - 9x\] Show Step 2

Next, replace all the \(x\)’s with \(y\)’s and all the original \(y\)’s with \(x\)’s.

\[x = 5 - 9y\] Show Step 3

Solve the equation from Step 2 for \(y\).

\[\begin{align*}x & = 5 - 9y\\ 9y & = 5 - x\\ y & = \frac{{5 - x}}{9}\end{align*}\] Show Step 4

Replace \(y\) with \({h^{ - 1}}\left( x \right)\).

\[{h^{ - 1}}\left( x \right) = \frac{{5 - x}}{9}\] Show Step 5

Finally, do a quick check by computing one or both of \(\left( {h \circ {h^{ - 1}}} \right)\left( x \right)\) and \(\left( {{h^{ - 1}} \circ h} \right)\left( x \right)\) and verify that each is \(x\). In general, we usually just check one of these and well do that here.

\[\left( {h \circ {h^{ - 1}}} \right)\left( x \right) = h\left[ {{h^{ - 1}}\left( x \right)} \right] = P\left[ {\frac{{5 - x}}{9}} \right] = 5 - 9\left( {\frac{{5 - x}}{9}} \right) = 5 - \left( {5 - x} \right) = x\]

The check works out so we know we did the work correctly and have inverse.