Problem 44 - Algebra and Trigonometry

Question

Find the horizontal asymptote, if there is one, of the graph of each rational function. $$ f(x)=\frac{-3 x+7}{5 x-2} $$

Step-by-Step Solution

Answer

The horizontal asymptote of the function \(f(x) = \frac{-3x + 7}{5x - 2}\) is \(y = -3/5\).

Step 1: Identifying the Coefficients of the Highest Power of x in the Numerator and Denominator

Look at the coefficients of the x terms in the numerator and denominator. In our function, the coefficient in the numerator is -3 (from -3x) and in the denominator is 5 (from 5x).

Step 2: Calculating the Ratio of the Coefficients

Calculate the ratio of the coefficient in the numerator to the coefficient in the denominator. In this case, the ratio is \(-3/5\)

Step 3: Identify the Horizontal Asymptote

The ratio obtained in step 2 gives the equation of the horizontal asymptote. Therefore, the line \(y = -3/5\) is the horizontal asymptote.