Question

complete the following for the piece - wise - defined function $f(x)=\begin{cases}6&\text{if }x < - 1\\|x|&\text{if }-1leq x < 3\\-6&\text{if }xgeq3end{cases}$. a. find $f(2)$, $f(-8)$, and $f(8)$. b. sketch the graph of $y = f(x)$. c. find the domain of $f$. d. find the range of $f$. a. $-6,6$ b. ${-6}cup1,3)cup{6}$ c. $(-infty,infty)$ d. ${-6}cup0,3)cup{6}$

Explanation:

Step1: Find f(2)

Since - 1≤2<3, use f(x)=|x|. So f(2)=|2| = 2.

Step2: Find f(-8)

Since - 8 < - 1, use f(x)=6. So f(-8)=6.

Step3: Find f(8)

Since 8≥3, use f(x)= - 6. So f(8)= - 6.

Step4: Sketch the graph

For x < - 1, y = 6 (a horizontal line at y = 6). For - 1≤x<3, y = |x|, which is a V - shaped graph with vertex at (0,0) in this interval. For x≥3, y=-6 (a horizontal line at y = - 6).

Step5: Find the domain

The function is defined for all real x. So the domain is (-∞,∞).

Step6: Find the range

When x < - 1, y = 6. When - 1≤x<3, 0≤y<3 (from y = |x|). When x≥3, y=-6. So the range is {-6}∪[0,3)∪{6}.

Answer:

a. f(2)=2, f(-8)=6, f(8)= - 6
b. Sketch as described above
c. (-∞,∞)
d. D. {-6}∪[0,3)∪{6}