Question

graphing a piecewise-defined function
which graph represents the piecewise-defined function $f(x)=\

$$\begin{cases}-x + 4, & 0 \\leq x < 3 \\\\6, & x \\geq 3\\end{cases}$$

$?

Explanation:

Step1: Analyze first piece ($0 \leq x < 3$)

The function is $f(x) = -x + 4$.

• At $x=0$: $f(0) = -0 + 4 = 4$ (closed dot, since $x=0$ is included)
• At $x=3$: $f(3) = -3 + 4 = 1$ (open dot, since $x=3$ is not included)

This is a line decreasing from $(0,4)$ to $(3,1)$.

Step2: Analyze second piece ($x \geq 3$)

The function is $f(x) = 6$.

• At $x=3$: $f(3)=6$ (closed dot, since $x=3$ is included)

This is a horizontal line starting at $(3,6)$ and extending right.

Step3: Match to the graph

Only the fourth graph has a decreasing line from $(0,4)$ (closed dot) to $(3,1)$ (open dot), and a horizontal line starting at $(3,6)$ (closed dot) going right.

Answer:

The fourth graph (bottom-right option: decreasing line with closed dot at (0,4), open dot at (3,1); horizontal line with closed dot at (3,6) extending right)