### Home > A2C > Chapter 9 > Lesson 9.1.1 > Problem9-12

9-12.

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 $\sqrt { \phantom{a} }$ 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 does $f ^{-1}$ to $x$: subtracts $3$ divides by $2$ $(\phantom{a} )^{2}$ adds $1$
1. Copy and complete the following table for $g^{−1}\left(x\right)$ if $g ( x ) = \frac { 1 } { 3 } ( x + 1 ) ^ { 2 } - 2$

 1st 2nd 3rd 4th What $g$ does to $x$: adds $1$ $(\phantom{a} )^{2}$ divides by $3$ subtracts $2$ What $g^{−1}$ does to $x$:

Look at the table above.

1. Write the equations for $f ^{−1}\left(x\right)$ and $g^{−1}\left(x\right)$.

$f^{-1}(x)=\left(\frac{x-3}{2}\right)^{2}+1$