Exercise 17 Page 222

To graph the given piecewise function, we should think about the graph of each individual piece of the function. Then we can combine the graphs on one coordinate plane.

$y=\text{-}3x-2$

First we will graph $y=\text{-}3x-2$ for the domain $x\le \text{-}1.$ This function has a slope of $\text{-}3$ and a $y\text{-}$intercept of $\text{-}2.$ Since the endpoint is included, this piece should end with a closed circle.

Looking at the graph, we can see that all of the possible $y\text{-}$values are greater than or equal to $1.$

$y=x+2$

Next, we will graph $y=x+2$ for the domain $x>\text{-}1.$ Since the endpoint is not included, we will end the piece with an open circle.

From the graph, we can see that all $y\text{-}$values that are greater than $1$ will be produced by this portion.

Combining the Pieces

Finally, we can combine the pieces onto to one coordinate plane.

Looking at the pieces together, we can see that there are no gaps in the possible values of $x.$ We can also see there are no gaps in the possible values of $y$ but that they are all greater than or equal to $1.$ We can use these facts to write the domain and range of the function.