Exercise 1 Page 229

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.

$y=2x+4$

First, we will graph $y=2x+4$ for the domain $x\le \text{-}1.$ This function has a slope of $2$ and a $y\text{-}$intercept of $4.$ 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 less than or equal to $2.$

$y=\frac{1}{3}x-1$

Next, we will graph $y=\frac{1}{3}x-1$ for the domain $x>\text{-}1.$ Since the endpoint is not included, we will end the piece with an open circle.

Looking at this graph, we can see that the minimum $y\text{-}$value is reached when $x=\text{-}1.$ To find the corresponding $y\text{-}$value, we will substitute $x$ for $\text{-}1$ in the equation of this piece. Therefore, all $y\text{-}$values that are greater than $\text{-}\frac{4}{3}$ will be produced by this portion.

Combining the Pieces

Finally, we can combine the pieces onto 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 $y.$ We can use these facts to write the domain and range of the function.