Question

sketch a graph of the piecewise defined function.
$f(x)=\

$$\begin{cases}1 - x & \\text{if }x < -7 \\\\ 4 & \\text{if }x \\geq -7\\end{cases}$$

$

Explanation:

Step1: Analyze \(y = 1 - x\) for \(x<-7\)

Find two points. When \(x=-8\), \(y=1-(-8)=9\). The function \(y = 1 - x\) has a slope \(m=-1\). Since \(x < - 7\), we draw an open - circle at \(x=-7\). Substitute \(x = - 7\) into \(y=1 - x\), \(y=1-(-7)=8\).

Step2: Analyze \(y = 4\) for \(x\geq-7\)

The function \(y = 4\) is a horizontal line. We draw a closed - circle at \(x=-7\) (because \(x=-7\) is included in this part of the piece - wise function) and the line \(y = 4\) for all \(x\) values greater than or equal to \(-7\).

Answer:

To sketch the graph:

• For \(y = 1 - x\) with \(x < - 7\): Plot the point \((-8,9)\) and use the slope \(m=-1\) to draw the line, ending with an open - circle at the point \((-7,8)\).
• For \(y = 4\) with \(x\geq-7\): Plot a closed - circle at the point \((-7,4)\) and draw the horizontal line \(y = 4\) for \(x\geq-7\).