Question

study guide: lc #1 piecewise directions: show all work. work must prove your answer. the only resources that you should be using are your class notes, practice, and/or nc test desmos (bit.ly/ncdesmos). if using desmos, show work by writing what was entered into desmos, sketch what was graphed and/or write what part of the graph was used to solve. what pages could help you complete the study guide?

1. evaluate $-f(2) + f(-2) - \frac{1}{2}f(-4)$.

graph of a piecewise function
answer:

1. select all true statement(s).

graph of a piecewise function
a. $\frac{1}{2}f(0) = 0$
b. $f(-5) + f(1) > f(4) + f(-4)$
c. $5f(-4) > 2f(3) + f(2)$
d. $f(2) = 2$
e. $f(2) = 3$

1. consider the two piecewise functions, $f(x)$ & $g(x)$. find $5f(5) - g(-3)$.

$f(x) = \

$$\begin{cases} 3x + 1 & \\text{for } x \\leq 0 \\\\ x^2 & \\text{for } x > 0 \\end{cases}$$

$
$g(x) = \

$$\begin{cases} -x & \\text{for } x < 2 \\\\ 2 & \\text{for } x \\geq 2 \\end{cases}$$

$
answer:

1. what is $-3f(6) - f(-3)$?

$f(x) = \

$$\begin{cases} 2x + 8, & x \\leq -2 \\\\ x^2 - 3, & -2 < x \\leq 3 \\\\ \\sqrt{x + 3}, & x > 3 \\end{cases}$$

$
answer:

1. evaluate $-f(2) - 4f(5)$.

$f(x) = \

$$\begin{cases} -|x|, & x \\leq 3 \\\\ x - 1, & x > 3 \\end{cases}$$

$

1. the graph is $f(x)$. evaluate $f(-2) + f(1)$.

graph of a piecewise function
answer:

Explanation:

Step1: Find $f(2), f(-2), f(-4)$

From the graph: $f(2)=4$, $f(-2)=0$, $f(-4)=-3$

Step2: Substitute into the expression

$-f(2)+f(-2)-\frac{1}{2}f(-4) = -4 + 0 - \frac{1}{2}(-3)$

Step3: Calculate the result

$-4 + 0 + \frac{3}{2} = -\frac{8}{2} + \frac{3}{2} = -\frac{5}{2}$

Step1: Read values from the graph

$f(0)=0$, $f(-5)=5$, $f(1)=1$, $f(4)=3$, $f(-4)=0$, $f(3)=3$, $f(2)=2$

Step2: Verify each option

A. $\frac{1}{2}f(0)=\frac{1}{2}(0)=0$ → True
B. $f(-5)+f(1)=5+1=6$; $f(4)+f(-4)=3+0=3$ → $6>3$ → True
C. $5f(-4)=5(0)=0$; $2f(3)+f(2)=2(3)+2=8$ → $0>8$ → False
D. $f(2)=2$ → True
E. $f(2)=3$ → False

Step1: Calculate $f(5)$

For $f(x)$, $5>0$, so $f(5)=5^2=25$

Step2: Calculate $g(-3)$

For $g(x)$, $-3<2$, so $g(-3)=-(-3)=3$

Step3: Substitute into the expression

$5f(5)-g(-3)=5(25)-3=125-3=122$

Step1: Calculate $f(6)$

For $f(x)$, $6>3$, so $f(6)=\sqrt{6+3}=\sqrt{9}=3$

Step2: Calculate $f(-3)$

For $f(x)$, $-3\leq-2$, so $f(-3)=2(-3)+8=-6+8=2$

Step3: Substitute into the expression

$-3f(6)-f(-3)=-3(3)-2=-9-2=-11$

Answer:

$-\frac{5}{2}$

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