Question

graph this function.
f(x)=\begin{cases}x - 4&\text{if }xleq4\\frac{1}{2}x - 6&\text{if }x>4end{cases}
select points on the graph to plot them. select point fill to change a point from closed to open.

Explanation:

Step1: Analyze \(y = x - 4\) for \(x\leq4\)

Find two - point form. When \(x = 0\), \(y=0 - 4=-4\); when \(x = 4\), \(y=4 - 4 = 0\). Plot \((0,-4)\) and \((4,0)\) with a closed - circle at \((4,0)\) since \(x = 4\) is included in this part of the domain, and draw a line through these two points for \(x\leq4\).

Step2: Analyze \(y=\frac{1}{2}x - 6\) for \(x>4\)

Find a point. When \(x = 4\), \(y=\frac{1}{2}\times4-6=2 - 6=-4\). But \(x = 4\) is not included in this part of the domain, so we have an open - circle at \((4,-4)\). When \(x = 6\), \(y=\frac{1}{2}\times6-6=3 - 6=-3\). Plot \((6,-3)\) and draw a line starting from the open - circle at \((4,-4)\) for \(x>4\).

Answer:

The graph consists of a line \(y = x - 4\) for \(x\leq4\) with endpoints \((0,-4)\) and \((4,0)\) (closed - circle at \((4,0)\)) and a line \(y=\frac{1}{2}x - 6\) for \(x>4\) with an open - circle at \((4,-4)\) and another point like \((6,-3)\) and the line extending to the right from the open - circle.