Question - Analysis of Function Domain, Asymptotes, and Behavior

The domain of \( f(x) \) is all real numbers except where the denominator equals zero. Set the denominator equal to zero and solve for x:

\( x^2 - 1 = 0 \)

\( (x + 1)(x - 1) = 0 \)

\( x = 1 \) or \( x = -1 \)

So the domain is \( x \in \mathbb{R} \), \( x \neq 1 \), \( x \neq -1 \).

To find vertical asymptotes, look at the points where the function is undefined, which are at \( x = 1 \) and \( x = -1 \).

To find the horizontal asymptote, examine the degrees of the numerator and denominator:

The degree of the numerator (2) is equal to the degree of the denominator (2). The horizontal asymptote is the ratio of the leading coefficients:

\( y = \frac{2}{1} = 2 \)

Analyze the behavior around the vertical asymptotes:

As \( x \) approaches \( 1 \) from the left, \( f(x) \) goes to \( -\infty \).

As \( x \) approaches \( 1 \) from the right, \( f(x) \) goes to \( +\infty \).

As \( x \) approaches \( -1 \) from the left, \( f(x) \) goes to \( +\infty \).

As \( x \) approaches \( -1 \) from the right, \( f(x) \) goes to \( -\infty \).

The function approaches the horizontal asymptote \( y = 2 \) as \( x \) goes to \( \pm\infty \).