Graphing Piecewise and Step Functions

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.

$f(x)=8$

First we will graph $f(x)=8$ for the domain $x\le \text{-}1.$ This function has a slope of $0.$ Since the endpoint is included, this piece should end with a closed circle.

Looking at the graph, we can see that the only possible value for $f(x)$ is $8.$

$f(x)=2x$

Next, we will graph $f(x)=2x$ for the domain $\text{-}1<x<4.$ This function has a slope of $2$ and intercepts the vertical axis at $0.$ Since the endpoints are not included, this piece should have open circles on both ends.

Looking at the graph, we can see that all the possible values for $f(x)$ are between $\text{-}2$ and $8.$

$f(x)=\text{-}4-x$

Then, we will graph $f(x)=\text{-}4-x$ for the domain $x\ge 4.$ This function has a slope of $\text{-}1.$ Since the endpoint is included, we will end the piece with a closed circle.

From the graph, we can see that all the values for $f(x)$ that are less than or equal to $\text{-}8$ will be produced by this piece.

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 $x.$ Conversely, the values that $f(x)$ takes are less than or equal to $\text{-}8,$ or greater than $\text{-}2$ and less than or equal to $8.$ We can use these facts to write the domain and range of the function. $$\begin{array}{rl}\text{Domain: }& \{\text{all real numbers}\}\\ \text{Range: }& \{y\le \text{-}8,\text{ and }\text{-}2<y\le 8\}\end{array}$$