Question

suppose that the function g is defined, for all real numbers, as follows.
g(x)=\begin{cases}-1&\text{if }x < 0\\-3&\text{if }x = 0\\3&\text{if }x>0end{cases}
graph the function g.

Explanation:

Step1: Analyze \(x < 0\)

For \(x<0\), \(g(x)= - 1\). This is a horizontal line \(y = - 1\) for all \(x\) - values less than \(0\). We use an open - circle at \(x = 0\) since \(x=0\) is not included in this part of the domain.

Step2: Analyze \(x = 0\)

When \(x = 0\), \(g(x)=-3\). So we plot the point \((0, - 3)\) as a closed - circle because \(x = 0\) is included in this part of the domain.

Step3: Analyze \(x>0\)

For \(x>0\), \(g(x)=3\). This is a horizontal line \(y = 3\) for all \(x\) - values greater than \(0\). We use an open - circle at \(x = 0\) since \(x = 0\) is not included in this part of the domain.

To graph:

1. Draw a horizontal line \(y=-1\) for \(x < 0\) with an open - circle at the point \((0,-1)\).
2. Plot the point \((0, - 3)\) as a closed - circle.
3. Draw a horizontal line \(y = 3\) for \(x>0\) with an open - circle at the point \((0,3)\).

Answer:

The graph consists of a horizontal line \(y=-1\) for \(x < 0\) (open - circle at \((0,-1)\)), the point \((0,-3)\) (closed - circle), and a horizontal line \(y = 3\) for \(x>0\) (open - circle at \((0,3)\)).