Question

1. graph each piecewise function.

a) $f(x)=\

$$\begin{cases}-x + 3, & x < 2 \\\\ \\frac{x}{2}, & x \\geq 2\\end{cases}$$

$

b) $f(x)=\

$$\begin{cases}-x + 1, & x < 0 \\\\ 1, & 0 \\leq x \\leq 2 \\\\ 3x - 5, & x > 2\\end{cases}$$

$

c) $f(x)=\

$$\begin{cases}-2x - 5, & x < -3 \\\\ x, & -3 \\leq x \\leq 2 \\\\ 4, & x > 2\\end{cases}$$

$

d) $f(x)=\

$$\begin{cases}x + 2, & x < -3 \\\\ -1, & -3 \\leq x < 0 \\\\ 2x, & 0 \\leq x < 2 \\\\ x - 5, & x \\geq 2\\end{cases}$$

$

Explanation:

Part a)

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

Function: $f(x) = -x + 3$

• At $x=2$, $f(2) = -2+3=1$ (open circle, since $x<2$)
• Y-intercept: at $x=0$, $f(0)=3$
• Plot line through $(0,3)$ with open circle at $(2,1)$

Step2: Analyze second piece ($x\geq2$)

Function: $f(x) = \frac{x}{2}$

• At $x=2$, $f(2)=\frac{2}{2}=1$ (closed circle)
• At $x=4$, $f(4)=\frac{4}{2}=2$
• Plot line through $(2,1)$ (closed) and $(4,2)$

Part b)

Step1: Analyze first piece ($x<0$)

Function: $f(x) = -x + 1$

• At $x=0$, $f(0)=-0+1=1$ (open circle)
• At $x=-2$, $f(-2)=2+1=3$
• Plot line through $(-2,3)$ with open circle at $(0,1)$

Step2: Analyze second piece ($0\leq x\leq2$)

Function: $f(x)=1$

• Horizontal line from closed circle $(0,1)$ to closed circle $(2,1)$

Step3: Analyze third piece ($x>2$)

Function: $f(x)=3x-5$

• At $x=2$, $f(2)=6-5=1$ (open circle)
• At $x=3$, $f(3)=9-5=4$
• Plot line through $(2,1)$ (open) and $(3,4)$

Part c)

Step1: Analyze first piece ($x<-3$)

Function: $f(x)=-2x-5$

• At $x=-3$, $f(-3)=6-5=1$ (open circle)
• At $x=-4$, $f(-4)=8-5=3$
• Plot line through $(-4,3)$ with open circle at $(-3,1)$

Step2: Analyze second piece ($-3\leq x\leq2$)

Function: $f(x)=x$

• Closed circle at $(-3,-3)$, closed circle at $(2,2)$
• Plot line through $(-3,-3)$ and $(2,2)$

Step3: Analyze third piece ($x>2$)

Function: $f(x)=4$

• Horizontal line starting with open circle at $(2,4)$, extending right

Part d)

Step1: Analyze first piece ($x<-3$)

Function: $f(x)=x+2$

• At $x=-3$, $f(-3)=-3+2=-1$ (open circle)
• At $x=-5$, $f(-5)=-5+2=-3$
• Plot line through $(-5,-3)$ with open circle at $(-3,-1)$

Step2: Analyze second piece ($-3\leq x<0$)

Function: $f(x)=-1$

• Horizontal line from closed circle $(-3,-1)$ to open circle $(0,-1)$

Step3: Analyze third piece ($0\leq x<2$)

Function: $f(x)=2x$

• At $x=0$, $f(0)=0$ (closed circle)
• At $x=2$, $f(2)=4$ (open circle)
• Plot line through $(0,0)$ and $(2,4)$ (open at $x=2$)

Step4: Analyze fourth piece ($x\geq2$)

Function: $f(x)=x-5$

• At $x=2$, $f(2)=2-5=-3$ (closed circle)
• At $x=4$, $f(4)=4-5=-1$
• Plot line through $(2,-3)$ (closed) and $(4,-1)$

Answer:

Graph Summary (Key Points for Plotting):

a)

• Line $y=-x+3$: $(0,3)$, open circle at $(2,1)$
• Line $y=\frac{x}{2}$: closed circle at $(2,1)$, $(4,2)$

b)

• Line $y=-x+1$: $(-2,3)$, open circle at $(0,1)$
• Horizontal line $y=1$: closed $(0,1)$ to closed $(2,1)$
• Line $y=3x-5$: open circle at $(2,1)$, $(3,4)$

c)

• Line $y=-2x-5$: $(-4,3)$, open circle at $(-3,1)$
• Line $y=x$: closed $(-3,-3)$ to closed $(2,2)$
• Horizontal line $y=4$: open circle at $(2,4)$, rightward

d)

• Line $y=x+2$: $(-5,-3)$, open circle at $(-3,-1)$
• Horizontal line $y=-1$: closed $(-3,-1)$ to open $(0,-1)$
• Line $y=2x$: closed $(0,0)$, open circle at $(2,4)$
• Line $y=x-5$: closed $(2,-3)$, $(4,-1)$

(Plot these points and lines on the provided coordinate grids as described.)