Exercise 32 Page 17

First, we will take a look at the given graph.

As we can see, each $y\text{-}$value of $f$ is shifted $1$ unit to the right and $3$ units down. To write the function rule $g(x)$ let's look at a table of transformations.

Transformations of $f(x)$
Horizontal Translations	$$\begin{array}{c}\text{Translation right }h\text{ units, }h>0\\ y=f(x-h)\end{array}$$
	$$\begin{array}{c}\text{Translation left }h\text{ units, }h>0\\ y=f(x+h)\end{array}$$
Vertical Translations	$$\begin{array}{c}\text{Translation up }k\text{ units, }k>0\\ y=f(x)+k\end{array}$$
	$$\begin{array}{c}\text{Translation down }k\text{ units, }k>0\\ y=f(x)-k\end{array}$$

Using the table, let's apply the first transformation, a horizontal translation $1$ unit to the right, by substituting $(x-1)$ for each $x\text{-}$term. Next, we will apply the second transformation, a translation $3$ units down, by subtracting $3$ from both sides of the equation. Finally, we can replace $f(x-1)-3$ with $g(x)$ and complete the function rule.