Question

f(x) = \

$$\begin{cases} \\frac{1}{2}x + 6, & -4 \\leq x < 0 \\\\ -5, & 0 \\leq x \\leq 7 \\end{cases}$$

what is the graph of f? choose 1 answer: a (graph with y-axis, x-axis, points and lines) b (graph with y-axis, x-axis, points and lines)

Explanation:

Step1: Find left segment endpoints

For $f(x)=\frac{1}{2}x+6$, $x=-4$: $\frac{1}{2}(-4)+6=4$ (closed dot at $(-4,4)$). $x=0$: $\frac{1}{2}(0)+6=6$ (open dot at $(0,6)$? No, wait: domain is $-4\leq x<0$, so $x=0$ is excluded (open dot), $x=-4$ included (closed dot).

Step2: Find right segment details

For $f(x)=-5$, domain $0\leq x\leq7$: closed dot at $(0,-5)$, closed dot at $(7,-5)$, horizontal line between.

Step3: Match to options

Option B has closed dot at $(-4,4)$, open dot at $(0,6)$ for left segment; closed dot at $(0,-5)$ and $(7,-5)$ for right segment, which matches.

Answer:

B. <The graph with a closed dot at (-4,4), open dot at (0,6) for the line segment of $y=\frac{1}{2}x+6$, and a horizontal line segment from closed dot (0,-5) to closed dot (7,-5)>