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.

$f(x)=\text{-}3x$

First we will graph $f(x)=\text{-}3x$ for the domain $x\le \text{-}4.$ This function has a slope of $\text{-}3$ and a $y$-intercept of $0.$ Since the endpoint is included, this piece should end with a closed circle.

Looking at the graph, we can see that all the possible $y$-values are greater than or equal to $12$ will be produced by this piece.

$f(x)=x$

Next, we will graph $f(x)=x$ for the domain $0<x\le 3.$ Since the endpoint $0$ is not included, we will display an open circle on this end. However, since the endpoint $3$ is included, we will display a closed circle on this end.

From the graph, we can see that all $y$-values that are greater than $0$ and less than or equal to $3$ will be produced by this piece.

$f(x)=8$

Then, we will graph $f(x)=8$ for the domain $x>3.$ Since the endpoint is not included, this will be an open circle.

From the graph, we can see that all $y$-values that are equal to $8$ 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 is a gap in the possible values of $x$ between $\text{-}4$ and $0.$ We can also see there are no $y$-values less than or equal to 0. Moreover, there are gaps in the possible values of $f(x)$ for $y$-values between $3$ and $8,$ as well as $8$ and $12.$ We can use these facts to write the domain and range of the function. $$\begin{array}{rl}\text{Domain: }& \{x\le \text{-}4\text{ and }x>0\}\\ \text{Range: }& \{0<y\le 3,y=8,\\ & \text{ and }y\ge 12\}\end{array}$$