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.

$f(x)=\text{-}2$

First we will graph $f(x)=\text{-}2$ for the domain $x<\text{-}4.$ This function has a slope of $0$ and a $y$-intercept of $\text{-}2.$ Since the endpoint is not included, this piece should end with an open circle.

Looking at the graph, we can see that the only possible $g\text{-}$value is equal to $\text{-}2.$

$f(x)=x-3$

Next, we will graph $f(x)=x-3$ for the domain $\text{-}1\le x\le 5.$ Since both endpoints are included, we will display them as closed circles.

From the graph, we can see that all the $y$-values that are between $\text{-}4$ and $2$ will be produced by this piece.

$f(x)=2x-15$

Then, we will graph $f(x)=2x-15$ for the domain $x>7.$ Since the endpoint is not included, we will display an open circle.

From the graph, we can see that all the $y$-values that are greater than $\text{-}1$ 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 gaps in the possible values of $x$ between $\text{-}4$ and $\text{-}1$, and between $5$ and $7.$ We can also see that all the possible values of $f(x)$ are greater than or equal to $\text{-}4,$ with no gaps. We can use these facts to write the domain and range of the function. $$\begin{array}{rl}\text{Domain: }& \{x<\text{-}4,\text{-}1\le x\le 5\text{ and }x>7\}\\ \text{Range: }& \{y\ge \text{-}4\}\end{array}$$