Question

sketch a graph of the piece - wise function.
$f(x)=\

$$\begin{cases}x + 1&\\text{if }x < - 2\\ - 2x-3&\\text{if }x\\geq - 2\\end{cases}$$

$

Explanation:

Step1: Analyze $y = x + 1$ for $x < - 2$

The slope is 1 and y - intercept is 1. When $x=-2$, $y=-2 + 1=-1$. Since $x < - 2$, we have an open - circle at the point $(-2,-1)$ and draw the line with slope 1 for $x$ values less than $-2$.

Step2: Analyze $y=-2x - 3$ for $x\geq - 2$

The slope is $-2$ and y - intercept is $-3$. When $x = - 2$, $y=-2\times(-2)-3=4 - 3 = 1$. We have a closed - circle at the point $(-2,1)$ and draw the line with slope $-2$ for $x$ values greater than or equal to $-2$.

The correct graph has an open - circle at $(-2,-1)$ for the line $y=x + 1$ (left part) and a closed - circle at $(-2,1)$ for the line $y=-2x - 3$ (right part).

Answer:

The graph with an open - circle at $(-2,-1)$ for $y=x + 1$ when $x < - 2$ and a closed - circle at $(-2,1)$ for $y=-2x - 3$ when $x\geq - 2$.