A table is a useful tool for writing some inverse functions. When the function has only one $x$ in its equation, 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 $g^{–1}\left(x\right)$ if $g\left(x\right)=\frac{1}{3}\left(x+1\right)^2-2$.

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

   Answer (a):

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

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

   Partial Answer (b):

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