Question

find the horizontal asymptote, if any, of the graph of the rational function.

$f(x)=\frac{-3x + 2}{5x + 7}$

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the horizontal asymptote is \square. (type an equation. simplify your answer. use integers or fractions for any numbers in the

b. there is no horizontal asymptote.

Explanation:

Step1: Determine the degrees of numerator and denominator

For the rational function \(f(x)=\frac{-3x + 2}{5x+7}\), the degree of the numerator \(n\) (degree of \(-3x+2\)) is \(n = 1\) (since the highest - power of \(x\) is \(x^1\)), and the degree of the denominator \(m\) (degree of \(5x + 7\)) is \(m=1\).

Step2: Use the rule for horizontal asymptotes of rational functions

When \(n=m\) (where \(n\) is the degree of the numerator and \(m\) is the degree of the denominator) for a rational function \(y=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}\), the horizontal asymptote is given by \(y=\frac{a_n}{b_m}\).
Here, \(a_n=-3\) (the leading coefficient of the numerator \(-3x + 2\)) and \(b_m = 5\) (the leading coefficient of the denominator \(5x+7\)).

Answer:

A. The horizontal asymptote is \(y=-\frac{3}{5}\)