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)=\

$$\begin{cases}-x + 3, & x < 2 \\\\ 3, & 2\\leq x < 4 \\\\ 4-2x, & x\\geq4\\end{cases}$$

$

Explanation:

Step 1: Graph \(y=-x + 3\) for \(x<2\)

For the linear function \(y=-x + 3\), when \(x = 2\), \(y=-2 + 3=1\). Since \(x<2\), the endpoint at \(x = 2\) is an open - circle. We can find two points on this line, for example, when \(x=0\), \(y = 3\) and when \(x=1\), \(y=2\), and then draw the line with an open - circle at \(x = 2\).

Step 2: Graph \(y = 3\) for \(2\leq x<4\)

This is a horizontal line. When \(x = 2\), since \(x\) can be equal to \(2\) in this part of the piece - wise function, the left - hand endpoint at \(x = 2\) is a closed - circle (because of the \(\leq\) sign). When \(x=4\), since \(x<4\) in this part, the right - hand endpoint at \(x = 4\) is an open - circle.

Step 3: Graph \(y=4-2x\) for \(x\geq4\)

For the linear function \(y = 4-2x\), when \(x = 4\), \(y=4-2\times4=-4\). Since \(x\geq4\), the endpoint at \(x = 4\) is a closed - circle. We can find another point, for example, when \(x=5\), \(y=4-2\times5=-6\), and then draw the line starting from the closed - circle at \(x = 4\).

Answer:

Graph \(y=-x + 3\) with an open - circle at \(x = 2\), \(y = 3\) with a closed - circle at \(x = 2\) and an open - circle at \(x = 4\), and \(y=4-2x\) with a closed - circle at \(x = 4\) as described above.