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# Graphing Piecewise and Step Functions

## Graphing Piecewise and Step Functions 1.6 - Solution

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.

### $f(x)=2x$

First we will graph $f(x)=2x$ for the domain $x\leq \text{-}6.$ This function has a slope of $2$ 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 less than or equal to $\text{-} 12.$

### $f(x)=5$

Next, we will graph $f(x)=5$ for the domain $\text{-} 6 Since the endpoint $\text{-}6$ is not included, we will display an open circle on this end. However, since the endpoint $2$ is included, we will display a closed circle on this end.

From the graph, we can see that all $y$-values that are equal to $5$ will be produced by this piece.

### $f(x)=\text{-} 2x+1$

Then, we will graph $f(x)=\text{-} 2x+1$ for the domain $x>4.$ Since the endpoint is not included, we will display an open circle on this end.

From the graph, we can see that all $y$-values that are less than $\text{-} 7$ 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 $2$ and $4.$ We can also see that the values for $f(x)$ are either $5$ or less than $\text{-} 7.$ We can use these facts to write the domain and range of the function.\begin{aligned} \textbf{Domain: }&\ \{x\leq 2 \text{ and } x>4\} \\ \textbf{Range: }&\ \{y < \text{-} 7 \text{ and } y=5\} \end{aligned}