graphing a piecewise defined function\nexplain how to graph the given piecewise - defined function. be sure…

Explanation:

Step1: Analyze $y=-x + 3, x<2$

For $y=-x + 3$, it's a linear - function. When $x = 2$, $y=-2 + 3=1$. Since $x<2$, the endpoint at $x = 2$ is an open - circle because the function is not defined at $x = 2$ for this part. We can find other points, for example, when $x=0$, $y = 3$. Plot the line $y=-x + 3$ for $x<2$ with an open - circle at $(2,1)$.

Step2: Analyze $y = 3,2\leq x<4$

This is a constant function. When $x = 2$, $y = 3$ and when $x=4$, $y = 3$. The left - hand endpoint at $x = 2$ is a closed - circle because $x$ can equal $2$ for this part, and the right - hand endpoint at $x = 4$ is an open - circle because $x$ cannot equal $4$ for this part. Plot the horizontal line $y = 3$ from $x = 2$ (closed - circle) to $x = 4$ (open - circle).

Step3: Analyze $y=4 - 2x,x\geq4$

For $y=4 - 2x$, when $x = 4$, $y=4-2\times4=-4$. Since $x\geq4$, the endpoint at $x = 4$ is a closed - circle. We can find other points, for example, when $x = 5$, $y=4-2\times5=-6$. Plot the line $y=4 - 2x$ for $x\geq4$ with a closed - circle at $(4,-4)$.

Answer:

Graph the line $y=-x + 3$ for $x<2$ with an open - circle at $(2,1)$, the horizontal line $y = 3$ for $2\leq x<4$ with a closed - circle at $(2,3)$ and an open - circle at $(4,3)$, and the line $y=4 - 2x$ for $x\geq4$ with a closed - circle at $(4,-4)$.