Question

which equation describes a rational function with x-intercepts at -4 and 2, a vertical asymptote at x = 1 and x = -1, and a horizontal asymptote at y = -3?

f(x) = \frac{-3(x-4)(x+2)}{x^2-1}

f(x) = \frac{x^2-1}{-3(x+4)(x-2)}

f(x) = \frac{x^2-1}{-3(x-4)(x+2)}

f(x) = \frac{-3(x+4)(x-2)}{x^2-1}

Explanation:

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<reasoning>

Determine the numerator from the x-intercepts

\[
\text{x-intercepts at } x = -4 \text{ and } x = 2 \implies \text{Numerator factors: } (x + 4)(x - 2)
\]

Determine the denominator from the vertical asymptotes

\[
\text{Vertical asymptotes at } x = 1 \text{ and } x = -1 \implies \text{Denominator factors: } (x - 1)(x + 1) = x^2 - 1
\]

Determine the leading coefficient from the horizontal asymptote

\[
\text{Horizontal asymptote at } y = -3 \implies \lim_{x \to \infty} f(x) = -3 \implies f(x) = \frac{-3(x + 4)(x - 2)}{x^2 - 1}
\]
</reasoning>

<answer>
<mcq-option>(A) \(f(x) = \frac{-3(x - 4)(x + 2)}{x^2 - 1}\)</mcq-option>
<mcq-option>(B) \(f(x) = \frac{x^2 - 1}{-3(x + 4)(x - 2)}\)</mcq-option>
<mcq-option>(C) \(f(x) = \frac{x^2 - 1}{-3(x - 4)(x + 2)}\)</mcq-option>
<mcq-correct>(D) \(f(x) = \frac{-3(x + 4)(x - 2)}{x^2 - 1}\)</mcq-correct>
</answer>

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Answer:

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<reasoning>

Determine the numerator from the x-intercepts

\[
\text{x-intercepts at } x = -4 \text{ and } x = 2 \implies \text{Numerator factors: } (x + 4)(x - 2)
\]

Determine the denominator from the vertical asymptotes

\[
\text{Vertical asymptotes at } x = 1 \text{ and } x = -1 \implies \text{Denominator factors: } (x - 1)(x + 1) = x^2 - 1
\]

Determine the leading coefficient from the horizontal asymptote

\[
\text{Horizontal asymptote at } y = -3 \implies \lim_{x \to \infty} f(x) = -3 \implies f(x) = \frac{-3(x + 4)(x - 2)}{x^2 - 1}
\]
</reasoning>

<answer>
<mcq-option>(A) \(f(x) = \frac{-3(x - 4)(x + 2)}{x^2 - 1}\)</mcq-option>
<mcq-option>(B) \(f(x) = \frac{x^2 - 1}{-3(x + 4)(x - 2)}\)</mcq-option>
<mcq-option>(C) \(f(x) = \frac{x^2 - 1}{-3(x - 4)(x + 2)}\)</mcq-option>
<mcq-correct>(D) \(f(x) = \frac{-3(x + 4)(x - 2)}{x^2 - 1}\)</mcq-correct>
</answer>

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