Question

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

$$\begin{cases} x - 1 & \\text{for} & x \\leq -5 \\\\ x - 10 & \\text{for} & x > 5 \\end{cases}$$

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

Explanation:

Step1: Find endpoint for $x\leq-5$

Calculate $f(-5) = -5 - 1 = -6$. This gives the closed endpoint $(-5, -6)$ (closed because $x\leq-5$ includes $x=-5$).

Step2: Plot second point for $x\leq-5$

Choose $x=-10$, $f(-10) = -10 - 1 = -11$. This gives the point $(-10, -11)$. Draw a line from $(-5, -6)$ through $(-10, -11)$, extending leftward (since $x$ goes to $-\infty$).

Step3: Find endpoint for $x>5$

Calculate $f(5) = 5 - 10 = -5$. This gives an open endpoint $(5, -5)$ (open because $x>5$ does not include $x=5$).

Step4: Plot second point for $x>5$

Choose $x=10$, $f(10) = 10 - 10 = 0$. This gives the point $(10, 0)$. Draw a line from $(5, -5)$ through $(10, 0)$, extending rightward (since $x$ goes to $+\infty$).

Answer:

• For the piece $f(x)=x-1$ ($x\leq-5$): A line with a closed endpoint at $(-5, -6)$, passing through $(-10, -11)$, extending infinitely left.
• For the piece $f(x)=x-10$ ($x>5$): A line with an open endpoint at $(5, -5)$, passing through $(10, 0)$, extending infinitely right.