Questions: Find the horizontal asymptote, if it exists, of the rational function below. If the function does not have a horizontal asymptote, enter NONE. g(x) = (4-x)(-4+4 x) / (4 x^2 + 1) The horizontal asymptote has equation

Find the horizontal asymptote, if it exists, of the rational function below. If the function does not have a horizontal asymptote, enter NONE.

g(x) = (4-x)(-4+4 x) / (4 x^2 + 1)

The horizontal asymptote has equation

Transcript text: Find the horizontal asymptote, if it exists, of the rational function below. If the function does not have a horizontal asymptote, enter NONE. \[ g(x)=\frac{(4-x)(-4+4 x)}{4 x^{2}+1} \] The horizontal asymptote has equation $\square$

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 degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Step 1: Identify the Degrees of the Polynomials

The given rational function is

\[ g(x) = \frac{(4-x)(4x-4)}{4x^2 + 1} \]

First, we expand the numerator:

\[ (4-x)(4x-4) = -4x^2 + 20x - 16 \]

The degree of the numerator is 2, and the degree of the denominator is also 2.

Step 2: Determine the Horizontal Asymptote

Since the degrees of the numerator and the denominator are equal, the horizontal asymptote is determined by the ratio of the leading coefficients of the numerator and the denominator.

The leading coefficient of the numerator $-4x^2$ is $-4$, and the leading coefficient of the denominator $4x^2$ is $4$.

Thus, the horizontal asymptote is:

\[ y = \frac{-4}{4} = -1 \]