Question - Analysis of Rational Function for Asymptotes and Domain

To determine the domain of f(x), find the x-values for which the denominator is non-zero:

\[ x^2 - 1 \neq 0 \]

\[ (x + 1)(x - 1) \neq 0 \]

\[ x \neq \pm1 \]

Thus, the domain of f(x) is all real numbers except x = -1 and x = 1.

Next, identify any vertical asymptotes by setting the denominator equal to zero:

\[ x^2 - 1 = 0 \]

\[ x = \pm1 \]

Therefore, there are vertical asymptotes at x = -1 and x = 1.

To find horizontal asymptotes, examine the degrees of the numerator and denominator. Since the degree of the numerator (2) is the same as the degree of the denominator (2), compute the ratio of the leading coefficients:

\[ \lim_{{x \to \pm\infty}} \frac{3x^2}{x^2-1} = 3 \]

There is a horizontal asymptote at y = 3.

Lastly, analyze the behavior of the function near the asymptotes. As x approaches 1 or -1 from the left or right, the function f(x) diverges to positive or negative infinity, depending on the direction of approach. The function approaches the horizontal asymptote y = 3 as x approaches plus or minus infinity.