Questions: What is the horizontal asymptote of the graph of f(x) = (x^2 + 3) / (2x^2 - 7x) ? Give your answer in the form y = a.

Transcript text: QUESTION 15 What is the horizontal asymptote of the graph of $f(x)=\frac{x^{2}+3}{2 x^{2}-7 x}$ ? Give your answer in the form $y=a$. Provide your answer below: FEEDBACK

Solution

Solution Steps

To find the horizontal asymptote of a rational function, we compare the degrees of the polynomial in the numerator and the polynomial in the denominator. If the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients.

Step 1: Identify the Function

We are given the function $ f(x) = \frac{x^{2} + 3}{2x^{2} - 7x} $.

Step 2: Determine the Degrees of the Polynomials

The degree of the polynomial in the numerator $ (x^{2} + 3) $ is 2, and the degree of the polynomial in the denominator $ (2x^{2} - 7x) $ is also 2.

Step 3: Calculate the Horizontal Asymptote

Since the degrees of the numerator and denominator are the same, the horizontal asymptote can be found by taking the ratio of the leading coefficients. The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is 2. Therefore, the horizontal asymptote is given by:

\[ y = \frac{1}{2} \]