Question

graph the piecewise linear function
$f(x)=\

$$\begin{cases}x - 5\\text{ if }x\\leq3\\\\-2\\text{ if }x > 3\\end{cases}$$

$
choose the correct graph.

Explanation:

Step1: Analyze $f(x)=x - 5$ for $x\leq3$

When $x = 3$, $f(3)=3 - 5=-2$. The slope of $y=x - 5$ is $m = 1$ and the $y$-intercept is $b=-5$. For $x\leq3$, we draw the line $y=x - 5$ including the point $(3,-2)$ (a closed - circle at $(3,-2)$ since $x$ can equal 3).

Step2: Analyze $f(x)=-2$ for $x>3$

For $x>3$, the function is a horizontal line $y = - 2$. We draw a horizontal line $y=-2$ starting from $x = 3$ but with an open - circle at $(3,-2)$ since $x$ cannot equal 3 for this part of the function.

Answer:

(Without seeing the actual graphs A, B, C, D, we can't give a specific letter - choice. But the correct graph should have a line $y=x - 5$ for $x\leq3$ with a closed - circle at $(3,-2)$ and a horizontal line $y=-2$ for $x>3$ with an open - circle at $(3,-2)$)