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8-12.

A table can be used as a useful tool for finding some inverse functions. When the function has only one x in it, the function can be described with a sequence of operations, each applied to the previous result. Consider the following table for $f(x)=2\sqrt{x-1}+3$.

 1st 2nd 3rd 4th What f does to x: subtracts $1$ $\sqrt {\hphantom{9}}$ multiplies by $2$ adds $3$

Since the inverse must undo these operations, in the opposite order, the table for $f ^{−1}\left(x\right)$ would look like the one below.

 1st 2nd 3rd 4th What $f^{−1}$ does to $x$: subtracts $3$ divides by $2$ $\left( \right)^{2}$ adds $1$
1. Copy and complete the following table for $g−1(x)\ \text{if}\ g(x)=\frac{1}{3}(x+1)^2−2\ \text{for}\ x\ge-1$.

 1st 2nd 3rd 4th What g does to x: adds $1$ $\left( \right)^{2}$ divides by $3$ subtracts $2$ What g−1 does to x: $\sqrt {\hphantom{9}}$

2. Write the equations for $f^{−1}\left(x\right)\ \text{and}\ g^{−1}\left(x\right)$.
$f^{-1}(x)=\left(\frac{x-3}{2}\right)^2+1$