Problem 43 - Algebra and Trigonometry

Question

Find the horizontal asymptote, if there is one, of the graph of each rational function. $$ f(x)=\frac{-2 x+1}{3 x+5} $$

Step-by-Step Solution

Answer

The horizontal asymptote of the function \(f(x) = \frac{-2x+1}{3x+5}\) is \(-2/3\).

Step 1: Identify the degrees of the polynomials

In the given function \(f(x)=\frac{-2x+1}{3x+5}\), the numerator and denominator are both polynomials of degree 1. This is because the highest power of x in both the numerator and denominator is 1.

Step 2: Determine the leading coefficients

The leading coefficient is the coefficient of the term with the highest degree. In the numerator, the leading coefficient is -2 and in the denominator, the leading coefficient is 3.

Step 3: Find the ratio of the leading coefficients

As the degrees are equal, the horizontal asymptote is found by taking the ratio of the leading coefficients. Therefore, the horizontal asymptote is \(-2/3\).