Question

which of the following graphs represents the given piecewise function?
f(x)=\begin{cases}\frac{1}{3}x + 6, & -6leq x < -3\\3, & -3leq x < 2\\-x + 7, & 2 < x < 7end{cases}

Explanation:

Step1: Analyze first - part of function

For $y = \frac{1}{3}x + 6$, when $x=-6$, $y=\frac{1}{3}\times(-6)+6=- 2 + 6=4$; when $x = - 3$, $y=\frac{1}{3}\times(-3)+6=-1 + 6 = 5$. It is a line segment with an open - circle at $x=-3$ and a closed - circle at $x = - 6$.

Step2: Analyze second - part of function

For $y = 3$ when $-3\leq x<2$, it is a horizontal line segment with a closed - circle at $x=-3$ and an open - circle at $x = 2$.

Step3: Analyze third - part of function

For $y=-x + 7$, when $x = 2$, $y=-2 + 7=5$; when $x = 7$, $y=-7 + 7=0$. It is a line segment with an open - circle at both $x = 2$ and $x = 7$.

Answer:

(No options are provided, but the above steps can be used to identify the correct graph among given options by checking the key - points and end - point types (open/closed circles) of each part of the piecewise function.)