Rational Functions and Zeros Practice Test

A rational function is a quotient of polynomials, $f(x)=\frac{p(x)}{q(x)}$. Zeros are the $x$-values where $f(x)=0$, which happens when $p(x)=0$ and $q(x)\neq 0$. Vertical asymptotes occur where $q(x)=0$ and the factor does not cancel. Consider

$$f(x)=\frac{x^2-1}{x-1}.$$

Factoring gives $x^2-1=(x-1)(x+1)$, so $(x-1)$ cancels for $x\neq 1$, leaving $f(x)=x+1$ with a hole at $x=1$. The remaining zero is where $x+1=0$, so $x=-1$ is a zero.

Using the function provided, what are the zeros of $f(x)=\frac{x^2-1}{x-1}$?

A rational function is a quotient of polynomials. For $f(x)=\dfrac{x^2-4}{x-2}$, zeros occur where $f(x)=0$ and the function is defined; vertical asymptotes come from non-canceling denominator zeros. Since $x^2-4=(x-2)(x+2)$, the function behaves like $x+2$ except at $x=2$, where it is undefined. Based on the function described, which statement best describes the behavior at its zero?