Question

suppose that the function g is defined, for all real numbers, as follows. g(x)=\begin{cases}1&\text{if }xleq - 2\\x&\text{if }-2 < xleq2\\1&\text{if }x > 2end{cases} graph the function g.

Explanation:

Step1: Analyze first - part of function

For \(g(x)=1\) when \(x\leq - 2\), we have a horizontal line \(y = 1\) with a closed - circle at \(x=-2\) (since \(x\) can equal \(-2\)).

Step2: Analyze second - part of function

For \(g(x)=x\) when \(-2 < x\leq2\), we find two points. When \(x=-2\), \(y=-2\) (open - circle as \(x
eq - 2\) for this part), and when \(x = 2\), \(y = 2\) (closed - circle as \(x\) can equal \(2\)). This is a straight - line segment with slope \(m = 1\).

Step3: Analyze third - part of function

For \(g(x)=1\) when \(x>2\), we have a horizontal line \(y = 1\) with an open - circle at \(x = 2\) (since \(x
eq2\) for this part).

Answer:

To graph:

1. Draw a horizontal line \(y = 1\) starting from \(x=-\infty\) and ending at \(x=-2\) with a closed - circle at \(x=-2\).
2. Draw the line \(y = x\) from the open - circle at the point \((-2,-2)\) to the closed - circle at the point \((2,2)\).
3. Draw a horizontal line \(y = 1\) starting from an open - circle at \(x = 2\) and extending to \(x=\infty\).