Question

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

$$\begin{cases} -6 & \\text{for} & x \\leq -1 \\\\ 2x - 18 & \\text{for} & x > 5 \\end{cases}$$

$

Explanation:

Step1: Analyze the first piece

The first piece of the function is \( f(x) = -6 \) for \( x \leq -1 \). This is a horizontal line. To graph it, we plot the point \( (-1, -6) \) (since \( x = -1 \) is included, we use a closed dot) and draw a horizontal line to the left (for all \( x \) values less than or equal to -1) at \( y = -6 \).

Step2: Analyze the second piece

The second piece is \( f(x) = 2x - 18 \) for \( x > 5 \). This is a linear function. First, find a point on this line. When \( x = 5 \), \( f(5) = 2(5) - 18 = 10 - 18 = -8 \), but since \( x > 5 \), we use an open dot at \( (5, -8) \). Then, we can find another point. For example, when \( x = 6 \), \( f(6) = 2(6) - 18 = 12 - 18 = -6 \). So we plot the open dot at \( (5, -8) \) and the point \( (6, -6) \), then draw a line with a slope of 2 (since the coefficient of \( x \) is 2) to the right (for all \( x \) values greater than 5).

Answer:

To graph the piecewise function:

• For \( x \leq -1 \), draw a horizontal line \( y = -6 \) with a closed dot at \( (-1, -6) \) and extend left.
• For \( x > 5 \), draw the line \( y = 2x - 18 \) with an open dot at \( (5, -8) \) and extend right, passing through points like \( (6, -6) \) (slope \( 2 \)).