Question

explain how to graph the given piecewise-defined function. be sure to specify the type of endpoint each piece of the function will have and why.

$f(x)=\

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

$

Explanation:

Step1: Analyze first piece $-x+3, x<2$

This is a linear function with slope $-1$ and y-intercept $3$. Since $x<2$ (strict inequality), calculate the endpoint at $x=2$: $f(2)=-2+3=1$. The endpoint at $(2,1)$ is an open circle because $x=2$ is not included in this piece's domain.

Step2: Graph first piece

Plot the y-intercept $(0,3)$, use the slope to find another point (e.g., $(1,2)$), then draw a line from these points to the open circle at $(2,1)$.

Step3: Analyze second piece $3, 2\leq x<4$

This is a horizontal line at $y=3$. For $x=2$ (included, $\leq$), the endpoint at $(2,3)$ is a closed circle. For $x=4$ (strict inequality $<$), the endpoint at $(4,3)$ is an open circle because $x=4$ is not in this piece's domain.

Step4: Graph second piece

Draw a horizontal line connecting the closed circle at $(2,3)$ to the open circle at $(4,3)$.

Step5: Analyze third piece $4-2x, x\geq4$

This is a linear function with slope $-2$ and y-intercept $4$. Calculate the endpoint at $x=4$: $f(4)=4-2(4)=-4$. Since $x\geq4$ (inclusive inequality), the endpoint at $(4,-4)$ is a closed circle because $x=4$ is included in this piece's domain.

Step6: Graph third piece

Use the slope to find another point (e.g., $(5,4-2(5))=-6$, so $(5,-6)$), then draw a line from the closed circle at $(4,-4)$ through this point, extending downward.

Answer:

1. For $f(x) = -x + 3$ where $x<2$:

• Graph the line with slope $-1$, y-intercept $(0,3)$. The endpoint at $x=2$ is $(2,1)$, drawn as an open circle because $x=2$ is not included in the domain of this piece.

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

• Graph a horizontal line at $y=3$. The left endpoint at $(2,3)$ is a closed circle (since $x=2$ is included) and the right endpoint at $(4,3)$ is an open circle (since $x=4$ is not included).

1. For $f(x) = 4-2x$ where $x\geq4$:

• Graph the line with slope $-2$. The endpoint at $x=4$ is $(4,-4)$, drawn as a closed circle because $x=4$ is included in the domain of this piece, then extend the line downward using the slope.