Question

watch the video and then solve the problem given below.
click here to watch the video.
the domain of the piecewise function is $(-\infty,\infty)$.
a. graph the function.
b. use your graph to determine the functions range.
$f(x)=\

$$\begin{cases}x + 5 & \\text{if } x < -2 \\\\ x - 5 & \\text{if } x \\geq -2\\end{cases}$$

$
a. choose the correct graph below.
\\(\bigcirc\\) a.\\(\bigcirc\\) b.\\(\bigcirc\\) c.\\(\bigcirc\\) d.

Explanation:

Step1: Analyze first piece ($x < -2$)

The first part is $f(x)=x+5$, a line with slope 1, y-intercept 5. At $x=-2$, $f(-2)=(-2)+5=3$, this is an open point (since $x<-2$). For $x<-2$, e.g., $x=-3$, $f(-3)=2$.

Step2: Analyze second piece ($x \geq -2$)

The second part is $f(x)=x-5$, a line with slope 1, y-intercept -5. At $x=-2$, $f(-2)=(-2)-5=-7$, this is a closed point (since $x\geq-2$). For $x>-2$, e.g., $x=0$, $f(0)=-5$.

Step3: Match to correct graph

Only Option D has an open point at $(-2,3)$ for $f(x)=x+5$ and closed point at $(-2,-7)$ for $f(x)=x-5$, with both lines having slope 1.

Step4: Find the range

For $x < -2$, $f(x)=x+5$ gives values $f(x) < 3$. For $x \geq -2$, $f(x)=x-5$ gives values $f(x) \geq -7$. Combining these, all real numbers are covered, so the range is $(-\infty, \infty)$.

Answer:

a. D. (The graph with open point at $(-2, 3)$ for the line $y=x+5$ and closed point at $(-2, -7)$ for the line $y=x-5$)
b. $(-\infty, \infty)$