CPM Homework Help : A2C Problem 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 $f ( x ) = 2 \sqrt { x - 1 } + 3$ .

	1^st	2^nd	3^rd	4^th
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.

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

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

Answer (a):

Look at the table above.

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