Question

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

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

$
click and drag to make a line. click the line to delete it.
click on an endpoint of a line to change it.

Explanation:

Step1: Analyze the first piece $f(x)=x + 6$ for $-4\leq x\lt - 1$

• Find two points on this line. When $x=-4$, $f(-4)=-4 + 6=2$. So the point is $(-4,2)$. When $x=-1$, $f(-1)=-1 + 6 = 5$, but since $x\lt - 1$ for the upper limit, the point $(-1,5)$ is an open circle.

Step2: Analyze the second piece $f(x)=6$ for $-1\lt x\lt6$

• This is a horizontal line. For $x=-1$, it's an open circle (since $x\gt - 1$) at $(-1,6)$ and for $x = 6$, it's an open circle at $(6,6)$ with $y = 6$ for all $x$ in $(-1,6)$.

Step3: Plot the points and draw the lines

• For the first line ($y=x + 6$), plot the point $(-4,2)$ (closed circle as $x=-4$ is included) and draw a line to $(-1,5)$ (open circle). For the second line ($y = 6$), draw a horizontal line from $(-1,6)$ (open circle) to $(6,6)$ (open circle) with $y = 6$ in between.

(Note: Since the problem is about graphing, the final answer here is the description of the graphing steps. If we were to represent the graph, we can't show it here but the steps above guide the graphing process.)

Answer:

To graph the piece - wise function:

1. For \(y=x + 6\) (\(-4\leq x\lt - 1\)):

• Plot the point \((-4,2)\) (closed circle).
• Draw a line from \((-4,2)\) to \((-1,5)\) (open circle at \((-1,5)\)).

1. For \(y = 6\) (\(-1\lt x\lt6\)):

• Draw a horizontal line from \((-1,6)\) (open circle) to \((6,6)\) (open circle) with \(y = 6\) for all \(x\) in the interval \((-1,6)\).