Question

Identify the asymptotes for the rational function: f(x)=x−1/x−4 (1 point) Responses x=4, y=1 x=4, y=1 x=1, y=4 x=1, y=4 x=-4, y=1 x=-4, y=1 x=1/4, y=0

Answer 1

To find the asymptotes of the function \( f(x) = \frac{x-1}{x-4} \), we need to determine the vertical and horizontal asymptotes.

### Vertical Asymptote:
Vertical asymptotes occur where the denominator is zero (and the numerator is not zero at this point).

For the function \( f(x) = \frac{x-1}{x-4} \), the denominator is zero when:
\[
x - 4 = 0 \implies x = 4
\]
Thus, there is a vertical asymptote at \( x = 4 \).

### Horizontal Asymptote:
To find the horizontal asymptote, we look at the degrees of the polynomial in the numerator and the denominator.
Both the numerator and denominator are degree 1 polynomials. For rational functions where the degree of the numerator equals the degree of the denominator, the horizontal asymptote is given by the ratio of the leading coefficients.

The leading coefficient of the numerator \( x - 1 \) is 1, and the leading coefficient of the denominator \( x - 4 \) is also 1. Therefore, the horizontal asymptote is:
\[
y = \frac{1}{1} = 1
\]

### Summary:
The function has:
- A vertical asymptote at \( x = 4 \)
- A horizontal asymptote at \( y = 1 \)

Thus, the correct response is:
**\( x=4, y=1 \)**