Question

which piecewise defined function is shown on the graph?
\\( f(x) = \

$$\begin{cases} x + 2, \\text{if } 0 < x < 4 \\\\ x - 3, \\text{if } 4 < x < 8 \\end{cases}$$

\\( f(x) = \

$$\begin{cases} x + 2, \\text{if } 0 \\leq x \\leq 4 \\\\ x - 3, \\text{if } 4 \\leq x \\leq 8 \\end{cases}$$

\\( f(x) = \

$$\begin{cases} x + 2, \\text{if } 0 < x \\leq 4 \\\\ x - 3, \\text{if } 4 < x \\leq 8 \\end{cases}$$

\\( f(x) = \

$$\begin{cases} x + 2, \\text{if } 0 \\leq x < 4 \\\\ x - 3, \\text{if } 4 \\leq x < 8 \\end{cases}$$

done

Explanation:

Step1: Analyze first segment domain

The left line starts at a closed dot at $x=0$ (so $x\geq0$) and ends at an open dot at $x=4$ (so $x<4$). Its equation is $y=x+2$ (when $x=0$, $y=2$; slope is 1).

Step2: Analyze second segment domain

The right line starts at a closed dot at $x=4$ (so $x\geq4$) and ends at an open dot at $x=8$ (so $x<8$). Its equation is $y=x-3$ (when $x=4$, $y=1$; slope is 1).

Step3: Match to function options

Combine the domains and equations to find the matching piecewise function.

Answer:

$\boldsymbol{f(x) =

$$\begin{cases} x+2,\text{if } 0 \leq x < 4 \\ x-3,\text{if } 4 \leq x < 8 \end{cases}$$

}$ (the fourth option)