AP Calculus AB Exam Question:

Consider the function $f(x) = \frac{x^2 - 4}{x - 2}$ defined for all real numbers except $x = 2$.

a) Determine the value of $f(2)$.

b) Determine the points of discontinuity, if any, for the function $f(x)$.

Solution:

a) To find the value of $f(2)$, we substitute $x = 2$ into the given function:

However, notice that we have a denominator of zero, which is undefined. Therefore, $f(2)$ is undefined.

b) We need to identify any potential points of discontinuity for the function $f(x)$.

The function has a vertical asymptote at $x = 2$ since we have a factor of $(x - 2)$ in the denominator.

To check for a removable discontinuity, we simplify the function by canceling out the common factor of $(x - 2)$:

By canceling out $(x - 2)$, we eliminated the point of discontinuity at $x = 2$, so there are no removable discontinuities.

Therefore, the only point of discontinuity for the function $f(x)$ is at $x = 2$, where we have a vertical asymptote.

Answer: a) $f(2)$ is undefined. b) The function has a point of discontinuity at $x = 2$ (vertical asymptote).