Question

which term below correctly completes the following sentence? if a function has a vertical asymptote at a certain x - value, then the function is ____ at that value. a. negative b. zero c. undefined d. rational

Explanation:

A vertical asymptote occurs at an \(x\) - value where the function is not defined. For example, in a rational function \(y=\frac{f(x)}{g(x)}\), vertical asymptotes are at the \(x\) - values where \(g(x) = 0\) (and \(f(x)
eq0\)). At these \(x\) - values, the function has a break (the function does not have a finite value), so it is undefined.

• Option A: A function can be negative at a point without having a vertical asymptote. For example, \(y=-x\) is negative for \(x>0\) but has no vertical asymptotes.
• Option B: A function is zero at a root (where \(y = 0\)), not at a vertical asymptote. For example, \(y=(x - 1)\) is zero at \(x = 1\) and has no vertical asymptote.
• Option D: A rational function is a type of function (\(y=\frac{f(x)}{g(x)}\), \(g(x)

eq0\)) but just being rational does not mean it has a vertical asymptote. A non - vertical line \(y=\frac{x+1}{1}\) is a rational function with no vertical asymptotes.

Answer:

C. undefined