Question

f(x) = \

$$\begin{cases} 5 - \\dfrac{1}{3}x, & -9 \\leq x < -3 \\\\ \\dfrac{2}{3}x, & -3 \\leq x \\leq 8 \\end{cases}$$

what is the graph of f?
choose 1 answer:
a graph a
b graph b

Explanation:

Step1: Find endpoints of first piece

For $f(x)=5-\frac{1}{3}x$, $x=-9$: $f(-9)=5-\frac{1}{3}(-9)=5+3=8$ (closed dot, since $x=-9$ is included).
$x=-3$: $f(-3)=5-\frac{1}{3}(-3)=5+1=6$ (open dot, since $x=-3$ is not included).

Step2: Find endpoints of second piece

For $f(x)=\frac{2}{3}x$, $x=-3$: $f(-3)=\frac{2}{3}(-3)=-2$ (closed dot, since $x=-3$ is included).
$x=8$: $f(8)=\frac{2}{3}(8)=\frac{16}{3}\approx5.33$ (closed dot, since $x=8$ is included).

Step3: Match to options

Compare calculated points to graphs: Option A has the first segment from $(-9,8)$ (closed) to $(-3,6)$ (open), and the second segment from $(-3,-2)$ (closed) to $(8,\frac{16}{3})$ (closed), which matches.

Answer:

A. <The graph with the first segment from (-9, 8) (closed dot) to (-3, 6) (open dot), and the second segment from (-3, -2) (closed dot) to (8, ~5.33) (closed dot)>