Question

which graph represents the function $f(x) = \frac{x^2 - 16}{x - 4}$?

Explanation:

Step1: Simplify the function

Factor the numerator:
$$f(x)=\frac{(x-4)(x+4)}{x-4}$$
Cancel $x-4$ (for $x
eq4$):

$$f(x)=x+4, \quad x eq4$$

Step2: Find domain restriction

The original function is undefined at $x=4$, so there is an open circle at $x=4$. Calculate $f(4)$ for the simplified line:
$$f(4)=4+4=8$$
So the open circle is at $(4,8)$.

Step3: Verify a valid point

Substitute $x=-4$ into the function:
$$f(-4)=\frac{(-4)^2-16}{-4-4}=\frac{16-16}{-8}=0$$
This gives the closed point $(-4,0)$, which lies on the line $y=x+4$.

Step4: Match to the correct graph

The graph that has a line $y=x+4$, an open circle at $(4,8)$, and a closed point at $(-4,0)$ is the top-right option.

Answer:

The top-right graph (with an open circle at (4, 8) and closed point at (-4, 0))