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Section 3.7 : Inverse Functions
2. Given \(\displaystyle g\left( x \right) = \frac{1}{2}x + 7\) find \({g^{ - 1}}\left( x \right)\) .
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Hint : Just follow the process outlines in the notes and you’ll be set to do this problem!
For the first step we simply replace the function with a \(y\).
\[y = \frac{1}{2}x + 7\] Show Step 2Next, replace all the \(x\)’s with \(y\)’s and all the original \(y\)’s with \(x\)’s.
\[x = \frac{1}{2}y + 7\] Show Step 3Solve the equation from Step 2 for \(y\).
\[\begin{align*}x & = \frac{1}{2}y + 7\\ 2x & = y + 14\\ 2x - 14 & = y\end{align*}\] Show Step 4Replace \(y\) with \({g^{ - 1}}\left( x \right)\).
\[{g^{ - 1}}\left( x \right) = 2x - 14\] Show Step 5Finally, do a quick check by computing one or both of \(\left( {g \circ {g^{ - 1}}} \right)\left( x \right)\) and \(\left( {{g^{ - 1}} \circ g} \right)\left( x \right)\) and verify that each is \(x\). In general, we usually just check one of these and well do that here.
\[\left( {g \circ {g^{ - 1}}} \right)\left( x \right) = g\left[ {{g^{ - 1}}\left( x \right)} \right] = g\left[ {2x - 14} \right] = \frac{1}{2}\left( {2x - 14} \right) + 7 = x - 7 + 7 = x\]The check works out so we know we did the work correctly and have inverse.