Final answer:
The correct rational function is \( f(x) = \frac{x + \frac{1}{2}}{(x - 2020)(x - 2022)} \), which matches option A. This function has vertical asymptotes at the correct x-values and a zero at x = -1/2. It will be continuous at x = 2021 as that value is not an asymptote or point of discontinuity.
Step-by-step explanation:
To write a rational function with vertical asymptotes at x = 2020 and x = 2022, and with a zero at x = -1/2, we look for a function of the form:
f(x) = \frac{N(x)}{(x - 2020)(x - 2022)}
where N(x) is the numerator that provides the specified zero. Now, given the options, the correct numerator to produce a zero at x = -1/2 is (x + 1/2). Therefore, the right choice is:
\( f(x) = \frac{x + \frac{1}{2}}{(x - 2020)(x - 2022)} \)
which corresponds to option A. Now, to determine if the function is continuous at the point x = 2021, we need to check if there are any points of discontinuity there. Since 2021 is not equal to the vertical asymptotes and is not where the function is undefined, the function will indeed be continuous at x = 2021.