Question

f(x) = \

$$\begin{cases} -x - 9 & \\text{for} & x \\leq -4 \\\\ x - 5 & \\text{for} & -4 < x \\leq 0 \\\\ 1 & \\text{for} & x > 0 \\end{cases}$$

line closed circle open circle
click and drag to make a line. click the line to delete it.
click on an endpoint of a line to change it.
graph with y-axis, x-axis, and partial graph drawn

Explanation:

To graph the piecewise function \( f(x) \), we analyze each piece:

Step 1: Graph \( f(x) = -x - 9 \) for \( x \leq -4 \)

• This is a linear function with slope \( -1 \) and y-intercept \( -9 \).
• When \( x = -4 \): \( f(-4) = -(-4) - 9 = 4 - 9 = -5 \). Plot a closed circle at \( (-4, -5) \) (since \( x \leq -4 \)).
• Choose another point, e.g., \( x = -10 \): \( f(-10) = -(-10) - 9 = 10 - 9 = 1 \). Plot \( (-10, 1) \) and draw a line through \( (-10, 1) \) and \( (-4, -5) \).

Step 2: Graph \( f(x) = x - 5 \) for \( -4 < x \leq 0 \)

• This is a linear function with slope \( 1 \) and y-intercept \( -5 \).
• When \( x = -4 \): \( f(-4) = -4 - 5 = -9 \). Plot an open circle at \( (-4, -9) \) (since \( x > -4 \)).
• When \( x = 0 \): \( f(0) = 0 - 5 = -5 \). Plot a closed circle at \( (0, -5) \).
• Draw a line through \( (-4, -9) \) (open circle) and \( (0, -5) \).

Step 3: Graph \( f(x) = 1 \) for \( x > 0 \)

• This is a horizontal line \( y = 1 \).
• When \( x = 0 \): Plot an open circle at \( (0, 1) \) (since \( x > 0 \)).
• Draw a horizontal line to the right from \( (0, 1) \) (open circle).

Final Graph Description:

• For \( x \leq -4 \): Line from \( (-\infty, \infty) \) through \( (-10, 1) \) to \( (-4, -5) \) (closed circle).
• For \( -4 < x \leq 0 \): Line from \( (-4, -9) \) (open circle) to \( (0, -5) \) (closed circle).
• For \( x > 0 \): Horizontal line \( y = 1 \) starting at \( (0, 1) \) (open circle) and extending right.

(Note: The graphing tool instructions suggest clicking and dragging to draw lines, adjusting endpoints with closed/open circles as needed.)

Answer:

To graph the piecewise function \( f(x) \), we analyze each piece:

Step 1: Graph \( f(x) = -x - 9 \) for \( x \leq -4 \)

• This is a linear function with slope \( -1 \) and y-intercept \( -9 \).
• When \( x = -4 \): \( f(-4) = -(-4) - 9 = 4 - 9 = -5 \). Plot a closed circle at \( (-4, -5) \) (since \( x \leq -4 \)).
• Choose another point, e.g., \( x = -10 \): \( f(-10) = -(-10) - 9 = 10 - 9 = 1 \). Plot \( (-10, 1) \) and draw a line through \( (-10, 1) \) and \( (-4, -5) \).

Step 2: Graph \( f(x) = x - 5 \) for \( -4 < x \leq 0 \)

• This is a linear function with slope \( 1 \) and y-intercept \( -5 \).
• When \( x = -4 \): \( f(-4) = -4 - 5 = -9 \). Plot an open circle at \( (-4, -9) \) (since \( x > -4 \)).
• When \( x = 0 \): \( f(0) = 0 - 5 = -5 \). Plot a closed circle at \( (0, -5) \).
• Draw a line through \( (-4, -9) \) (open circle) and \( (0, -5) \).

Step 3: Graph \( f(x) = 1 \) for \( x > 0 \)

• This is a horizontal line \( y = 1 \).
• When \( x = 0 \): Plot an open circle at \( (0, 1) \) (since \( x > 0 \)).
• Draw a horizontal line to the right from \( (0, 1) \) (open circle).

Final Graph Description:

• For \( x \leq -4 \): Line from \( (-\infty, \infty) \) through \( (-10, 1) \) to \( (-4, -5) \) (closed circle).
• For \( -4 < x \leq 0 \): Line from \( (-4, -9) \) (open circle) to \( (0, -5) \) (closed circle).
• For \( x > 0 \): Horizontal line \( y = 1 \) starting at \( (0, 1) \) (open circle) and extending right.

(Note: The graphing tool instructions suggest clicking and dragging to draw lines, adjusting endpoints with closed/open circles as needed.)