Question

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

$$\begin{cases} -x - 4 & \\text{for} & x < 1 \\\\ x - 13 & \\text{for} & x > 5 \\end{cases}$$

$
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:

Step1: Analyze \( f(x) = -x - 4 \) ( \( x < 1 \) )

This is a linear function with slope \( -1 \) and y - intercept \( -4 \). To graph it, we can find a point. When \( x = 0 \) (which is less than 1), \( f(0)=-0 - 4=-4 \). Also, as \( x \) approaches 1 from the left, \( f(1^{-})=-1 - 4=-5 \). So we draw a line with slope - 1, passing through (0, - 4) and approaching (1, - 5) (with an open circle at \( x = 1 \) since \( x<1 \)).

Step2: Analyze \( f(x)=x - 13 \) ( \( x > 5 \) )

This is a linear function with slope \( 1 \). When \( x = 6 \) (which is greater than 5), \( f(6)=6 - 13=-7 \). As \( x \) approaches 5 from the right, \( f(5^{+})=5 - 13=-8 \). So we draw a line with slope 1, passing through (6, - 7) and approaching (5, - 8) (with an open circle at \( x = 5 \) since \( x > 5 \)).

Answer:

To graph the piece - wise function:

1. For \( y=-x - 4, x < 1 \): Draw a line with slope - 1. Plot the point (0, - 4) and draw the line towards the left and towards \( x = 1 \) (open circle at \( x = 1,y=-5 \)).
2. For \( y=x - 13, x > 5 \): Draw a line with slope 1. Plot the point (6, - 7) and draw the line towards the right and towards \( x = 5 \) (open circle at \( x = 5,y=-8 \)).

(Note: Since this is a graphing problem, the final answer is the graphical representation as described above. If we were to describe the key points:

• For \( y=-x - 4(x < 1) \): passes through (0, - 4), has an open end at (1, - 5).
• For \( y=x - 13(x > 5) \): passes through (6, - 7), has an open end at (5, - 8))