Question

find equations for the horizontal asymptotes, if any, for the following rational function.
$f(x) = \frac{-3}{x - 9}$
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Explanation:

Step1: Recall Horizontal Asymptote Rules

For a rational function \( f(x) = \frac{N(x)}{D(x)} \), where \( N(x) \) is the numerator and \( D(x) \) is the denominator:

• If the degree of \( N(x) \) is less than the degree of \( D(x) \), the horizontal asymptote is \( y = 0 \).
• If the degrees are equal, the horizontal asymptote is \( y = \frac{\text{leading coefficient of } N(x)}{\text{leading coefficient of } D(x)} \).
• If the degree of \( N(x) \) is greater than the degree of \( D(x) \), there is no horizontal asymptote (but there may be an oblique asymptote).

Step2: Analyze the Given Function

For \( f(x) = \frac{-3}{x - 9} \), the numerator \( N(x) = -3 \) is a constant (degree 0), and the denominator \( D(x) = x - 9 \) is a polynomial of degree 1.
Since the degree of the numerator (0) is less than the degree of the denominator (1), by the rule above, the horizontal asymptote is \( y = 0 \).

Answer:

\( y = 0 \)