Question

graph the function.
$f(x)=\

$$\begin{cases}3x&\\text{for }x < -1\\\\-3x&\\text{for }x\\geq -1\\end{cases}$$

$

Explanation:

Step1: Analyze the first part of the piece - wise function ($x < - 1$)

The function is \(y = 3x\). When \(x=-2\), \(y = 3\times(-2)=-6\). When \(x=-1\) (but not included in this part), \(y = 3\times(-1)=-3\). This is a straight - line with a slope \(m = 3\). We draw an open circle at the point \((-1,-3)\) since \(x=-1\) is not in the domain \(x < - 1\) for this part of the function.

Step2: Analyze the second part of the piece - wise function ($x\geq - 1$)

The function is \(y=-3x\). When \(x=-1\), \(y=-3\times(-1) = 3\). When \(x = 0\), \(y=-3\times0=0\). When \(x = 1\), \(y=-3\times1=-3\). This is a straight - line with a slope \(m=-3\). We draw a closed circle at the point \((-1,3)\) since \(x = - 1\) is included in the domain \(x\geq - 1\) for this part of the function.

Answer:

To graph the function \(f(x)=

$$\begin{cases}3x, &x < - 1\\-3x, &x\geq - 1\end{cases}$$

\), draw the line \(y = 3x\) for \(x < - 1\) (with an open circle at \((-1,-3)\)) and the line \(y=-3x\) for \(x\geq - 1\) (with a closed circle at \((-1,3)\)).