Question

graph the piece - wise function shown below using line segments, rays, open circles and determine its domain and range: $f(x)=\begin{cases}x + 1&\text{if }xleq - 5\\6&\text{if }-4lt xleq - 2\\-2x - 1&\text{if }xgt1end{cases}$ be careful to include all features of $f(x)$ in your graph.

Explanation:

Step1: Graph $y = x + 1$ for $x\leq - 5$

Find the point at $x=-5$, $y=-5 + 1=-4$. The line has a slope of 1 and y - intercept of 1. Draw a ray starting at the point $(-5,-4)$ (closed - circle since $x = - 5$ is included) and going to the left.

Step2: Graph $y = 6$ for $-4\lt x\leq - 2$

Draw a horizontal line segment from the open - circle at $x=-4,y = 6$ to the closed - circle at $x=-2,y = 6$.

Step3: Graph $y=-2x - 1$ for $x>1$

Find the point at $x = 1$, $y=-2\times1-1=-3$. But $x = 1$ is not included. For other points, if $x = 2$, $y=-2\times2-1=-5$. Draw a ray starting at the open - circle at $(1,-3)$ and going to the right.

Step4: Determine the domain

The domain is the set of all possible $x$ values. Here, all real numbers are covered, so the domain is $(-\infty,\infty)$.

Step5: Determine the range

For $y=x + 1,x\leq - 5$, $y\leq - 4$. For $y = 6,-4\lt x\leq - 2$, $y = 6$. For $y=-2x - 1,x>1$, $y<-3$. So the range is $(-\infty,-3)\cup\{6\}$.

Answer:

Domain: $(-\infty,\infty)$; Range: $(-\infty,-3)\cup\{6\}$