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# Graphing Piecewise and Step Functions

## Graphing Piecewise and Step Functions 1.9 - 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.

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

First we will graph $f(x)=2x+2$ for the domain $x<\text{-}6.$ This function has a slope of $2$ and a $y$-intercept of $2.$ Since the endpoint is not included, this piece should end with an open circle.

Looking at the graph, we can see that all the possible $y$-values are less than $\text{-} 10.$

### $f(x)=x$

Next, we will graph $f(x)=x$ for the domain $\text{-} 6 \leq x\leq2.$ Since the endpoints are included, this piece should end with a closed circle on both ends.

From the graph, we can see that all $y$-values that are between $\text{-} 6$ and $2$ will be produced by this piece.

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

Then, we will graph $f(x)=\text{-} 3$ for the domain $x>2.$ Since the endpoint is not included, we will end the piece with an open circle.

From the graph, we can see that all values for $f(x)$ that are equal to $\text{-} 3$ 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 $x.$ However, we see that the $y\text{-}$variable takes values which are less than $\text{-} 10$ or between $\text{-} 6$ and $2.$ We can use these facts to write the domain and range of the function. \begin{aligned} \textbf{Domain: }&\ \{\text{all real numbers}\}\\ \textbf{Range: }&\ \{y<\text{-} 10 \text{ and } \text{-} 6\leq y \leq 2\} \end{aligned}