Question

sketch a graph of $f(x)=\begin{cases}-2&\text{if }xleq - 1\\x - 1&\text{if }-12end{cases}$

Explanation:

Step1: Graph $y = - 2$ for $x\leq - 1$

Draw a horizontal line $y=-2$ starting from $x = -\infty$ and ending at $x=-1$ with a closed - circle at $x = - 1$ since the inequality is $\leq$.

Step2: Graph $y=x - 1$ for $-1\lt x\leq2$

First, find the $y$ - values at the endpoints. When $x=-1$, $y=-1 - 1=-2$ (open - circle since $x\gt - 1$). When $x = 2$, $y=2 - 1=1$ (closed - circle since $x\leq2$). Then draw the line $y=x - 1$ between these two points.

Step3: Graph $y = 0$ for $x\gt2$

Draw a horizontal line $y = 0$ starting from $x = 2$ (open - circle since $x\gt2$) and extending to $x=\infty$.

Answer:

The graph consists of a horizontal line $y=-2$ for $x\leq - 1$ (closed - circle at $x=-1$), a line $y=x - 1$ for $-1\lt x\leq2$ (open - circle at $x=-1$, closed - circle at $x = 2$), and a horizontal line $y = 0$ for $x\gt2$ (open - circle at $x = 2$).