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 $f(x)=2\sqrt{x-1}+3$.

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.

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 $x$:			$\sqrt {\hphantom{9}}$

   Answer (a):

   1st: adds $2$
   2nd: multiplies by $3$
   4th: subtracts $1$

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

   Partial Answer (b):

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

   What is the correct domain for this inverse function?