Question

select the correct answer.

which statement describes the graph of the function \\(f(x) = \frac{x^2 - 1}{x^2 - 2x + 1}\\)?

• there is a hole at \\(x = -1\\).
• there is a vertical asymptote at \\(x = -1\\).
• the \\(y\\)-intercept is \\(y = -1\\).
• there is a horizontal asymptote at \\(y = 1\\).

Explanation:

Factor the numerator and denominator

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

Simplify the rational function and find domain restrictions

$$
LATEXBLOCK0
$$

Analyze asymptotes, holes, and intercepts

$$
LATEXBLOCK1
$$

Answer:

• There is a hole at \(x = -1\).
• There is a vertical asymptote at \(x = -1\).
• The \(y\)-intercept is \(y = -1\). (Correct answer)
• There is a horizontal asymptote at \(y = -1\).