Question

graph the following function on the axes provided.
$f(x) = \

$$\begin{cases} -6 & \\text{for} & x < 1 \\\\ x - 6 & \\text{for} & x > 4 \\end{cases}$$

$
click and drag to make a line. click the line to delete it.
click on an endpoint of a line to change it.
graph axes with x from -10 to 10 and y from -10 to 10

Explanation:

Step 1: Analyze the first part of the piece - wise function

For \(y = - 6\) when \(x<1\). This is a horizontal line. Since \(x < 1\), we draw an open - circle at \(x = 1,y=-6\) (because the function is not defined at \(x = 1\) for this part) and draw a horizontal line to the left of \(x = 1\) at \(y=-6\).

Step 2: Analyze the second part of the piece - wise function

For \(y=x - 6\) when \(x>4\). First, find the value of the function at \(x = 4\). Substitute \(x = 4\) into \(y=x - 6\), we get \(y=4-6=-2\). Since \(x>4\), we draw an open - circle at \(x = 4,y=-2\).
The function \(y=x - 6\) has a slope \(m = 1\) (the coefficient of \(x\)) and a \(y\) - intercept \(b=-6\). We can use the slope - intercept form. For \(x>4\), we can find another point. For example, when \(x=5\), \(y=5 - 6=-1\). We draw a line with slope \(1\) starting from the open - circle at \((4,-2)\) and going to the right.

Answer:

The graph consists of a horizontal line \(y = - 6\) for \(x<1\) (with an open - circle at \((1,-6)\)) and a line \(y=x - 6\) for \(x>4\) (with an open - circle at \((4,-2)\)).