Question

piecewise function
use technology to graph the function, then answer the following questions.
$f(x)=\

$$\begin{cases} -x-3, & x < -1 \\\\ \\sqrt{x+5}-4, & -1 \\leq x < 4 \\\\ (x-4)^2-1, & 4 \\leq x \\leq 5 \\end{cases}$$

$
11 multiple choice 2 points
is the function continuous or discontinuous?
○ continuous
○ discontinuous
12 numeric 2 points
find $f(4)$.
answer

Explanation:

Step1: Check continuity at $x=-1$

First, calculate left-hand limit: $\lim_{x \to -1^-} f(x) = -(-1) - 3 = -2$
Calculate right-hand limit: $\lim_{x \to -1^+} f(x) = \sqrt{-1+5} - 4 = 2 - 4 = -2$
Calculate $f(-1) = \sqrt{-1+5} - 4 = -2$. All values match.

Step2: Check continuity at $x=4$

First, calculate left-hand limit: $\lim_{x \to 4^-} f(x) = \sqrt{4+5} - 4 = 3 - 4 = -1$
Calculate right-hand limit: $\lim_{x \to 4^+} f(x) = (4-4)^2 - 1 = -1$
Calculate $f(4) = (4-4)^2 - 1 = -1$. All values match.

Step3: Evaluate $f(4)$

Use the third piece of the function since $4$ is in $4 \leq x \leq 5$:
$f(4) = (4-4)^2 - 1$

Answer:

For Question 11:

Continuous

For Question 12:

$-1$