Question

consider the following rational function.\\( f(x) = \frac{-2x - 5}{x - 4} \\)\
step 2 of 3: find equations for the horizontal or oblique asymptotes, if any, for the function.\
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Explanation:

Step1: Analyze degrees of numerator and denominator

For the rational function \( f(x)=\frac{-2x - 5}{x - 4} \), the degree of the numerator (highest power of \( x \)) is \( 1 \) (from \( -2x \)) and the degree of the denominator is also \( 1 \) (from \( x \)).

Step2: Find horizontal asymptote (since degrees are equal)

When the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients. The leading coefficient of the numerator is \( -2 \) and of the denominator is \( 1 \). So the horizontal asymptote is \( y=\frac{-2}{1}=-2 \).

Step3: Check for oblique asymptote

Oblique asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. Here, degrees are equal, so there is no oblique asymptote.

Answer:

\( y = -2 \)