Question

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

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

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

a.
the horizontal asymptote is ( y = -\frac{3}{7} ).
(type an equation. simplify your answer. use integers or fractions for any numbers in the equation.)
b. there is no horizontal asymptote.

Explanation:

Step1: Recall Horizontal Asymptote Rule

For a rational function \( f(x)=\frac{N(x)}{D(x)} \), where \( N(x) \) and \( D(x) \) are polynomials, if the degrees of \( N(x) \) and \( D(x) \) are equal (both degree 1 here, since \( N(x)=-3x + 5 \) and \( D(x)=7x + 2 \) are linear), the horizontal asymptote is \( y=\frac{\text{leading coefficient of } N(x)}{\text{leading coefficient of } D(x)} \).

Step2: Identify Leading Coefficients

Leading coefficient of \( N(x)=-3x + 5 \) is \( -3 \).
Leading coefficient of \( D(x)=7x + 2 \) is \( 7 \).

Step3: Calculate Horizontal Asymptote

Using the rule, \( y=\frac{-3}{7}=-\frac{3}{7} \).

Answer:

A. The horizontal asymptote is \( y = -\frac{3}{7} \)