Question

graphing a piecewise defined function
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

For $y=-x + 3$ with $x<2$. Find two points. When $x = 0$, $y=3$; when $x = 2$, $y=-2 + 3=1$. Since $x<2$, the endpoint at $x = 2$ is an open - circle (hollow dot) because the function is not defined at $x = 2$ for this piece.

Step2: Analyze second - piece

For $y = 3$ with $2\leq x<4$. It is a horizontal line. The left - hand endpoint at $x = 2$ is a closed - circle (solid dot) because the inequality includes $x = 2$, and the right - hand endpoint at $x = 4$ is an open - circle because $x<4$ for this piece.

Step3: Analyze third - piece

For $y=4 - 2x$ with $x\geq4$. When $x = 4$, $y=4-2\times4=-4$. Since $x\geq4$, the endpoint at $x = 4$ is a closed - circle. Find another point, e.g., when $x = 5$, $y=4-2\times5=-6$.

Step4: Graph the pieces

Plot the points and draw the lines for each piece according to the above - determined endpoints.

Answer:

To graph the piece - wise function: For $y=-x + 3$ with $x<2$, plot points and draw a line with an open - circle at $x = 2$. For $y = 3$ with $2\leq x<4$, draw a horizontal line with a closed - circle at $x = 2$ and an open - circle at $x = 4$. For $y=4 - 2x$ with $x\geq4$, draw a line with a closed - circle at $x = 4$.