Question

what is the domain of the function shown in the graph below?

Explanation:

Step1: Identify domain definition

The domain of a function is the set of all valid $x$-values for which the function is defined (i.e., where the graph exists).

Step2: Analyze left segment of graph

The left curve exists for $x$-values less than $-4$, up to but not including $x=-8$ (open circle at $x=-8$ means it is excluded). This interval is $(-\infty, -8) \cup (-4, \infty)$.

Step3: Analyze right segments of graph

The middle curve exists for $x$-values greater than $-4$ (open circle at $x=-4$ means it is excluded), and the right line extends infinitely to the right.

Step4: Combine valid intervals

Combine the intervals from the left and right segments: $(-\infty, -8) \cup (-4, \infty)$.

Answer:

The domain of the function is $(-\infty, -8) \cup (-4, \infty)$