Question

$f(x)=\

$$\begin{cases} -x - 10 & \\text{for} & x < -4 \\\\ 2x + 3 & \\text{for} & -4 \\leq x < 1 \\\\ -x + 6 & \\text{for} & x \\geq 1 \\end{cases}$$

$
line closed circle open circle
click and drag to make a line. click the line to delete it.
click on an endpoint of a line to change it.

Explanation:

Step1: Plot first piece endpoints

First, for $f(x) = -x -10$ where $x < -4$:

• At $x=-4$ (open circle, since $x<-4$): $f(-4) = -(-4)-10 = 4-10=-6$
• Choose another point, e.g., $x=-10$: $f(-10) = -(-10)-10=10-10=0$

Draw a line from $(-10,0)$ to open circle at $(-4,-6)$.

Step2: Plot second piece endpoints

For $f(x)=2x+3$ where $-4 \leq x <1$:

• At $x=-4$ (closed circle): $f(-4)=2(-4)+3=-8+3=-5$
• At $x=1$ (open circle): $f(1)=2(1)+3=2+3=5$

Draw a line from closed circle at $(-4,-5)$ to open circle at $(1,5)$.

Step3: Plot third piece endpoints

For $f(x)=-x+6$ where $x \geq1$:

• At $x=1$ (closed circle): $f(1)=-1+6=5$
• Choose another point, e.g., $x=10$: $f(10)=-10+6=-4$

Draw a line from closed circle at $(1,5)$ to $(10,-4)$.

Answer:

The piecewise function is graphed as follows:

1. A line from $(-10, 0)$ to an open circle at $(-4, -6)$ for $f(x)=-x-10$ ($x<-4$)
2. A line from a closed circle at $(-4, -5)$ to an open circle at $(1, 5)$ for $f(x)=2x+3$ ($-4\leq x<1$)
3. A line from a closed circle at $(1, 5)$ to $(10, -4)$ for $f(x)=-x+6$ ($x\geq1$)