### Home > INT3 > Chapter 8 > Lesson 8.1.1 > Problem8-10

8-10.

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

 $1$st $2$nd $3$rd $4$th What $f$ does to $x$: subtracts 1 $\sqrt { }$ 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 $f ^{–1}$ does to $x$: subtracts $3$ divides by $2$ $\left( \ \ \ \right)^{2}$ adds $1$
1. Copy and complete the following table for $g^{–1}\left(x\right)$ 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 $x$: $\sqrt { }$

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

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

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