Question

Identify the asymptotes for the rational function: f(x) = x−1x−4
(1 point)
Responses

Vertical asymptote: x = 4
, horizontal asymptote y=1

Vertical asymptote: x = 4
, horizontal asymptote y is equal to 1

Vertical asymptote x = 1
, horizontal asymptote y=4

Vertical asymptote x = 1
, horizontal asymptote y is equal to 4

Vertical asymptote x = −4
, horizontal asymptote y=1

Vertical asymptote x = −4
, horizontal asymptote y is equal to 1

Vertical asymptote x = 14
, horizontal asymptote y=0

Answer 1

To identify the asymptotes of the rational function \( f(x) = \frac{x - 1}{x - 4} \), we analyze both vertical and horizontal asymptotes.

1. **Vertical Asymptote**: Vertical asymptotes occur where the denominator of the function is equal to zero (and the numerator isn't also zero at those points).

For the function \( f(x) = \frac{x - 1}{x - 4} \):
- Set the denominator equal to zero: \( x - 4 = 0 \) leads to \( x = 4 \).

So there is a vertical asymptote at \( x = 4 \).

2. **Horizontal Asymptote**: Horizontal asymptotes are determined by comparing the degrees of the numerator and the denominator.

In our function:
- The degree of the numerator (1) is equal to the degree of the denominator (1).
- For rational functions where the degrees are equal, the horizontal asymptote is given by the ratio of the leading coefficients.

In this case:
- The leading coefficient of the numerator (1) is \( 1 \) (coefficient of \( x \)).
- The leading coefficient of the denominator (1) is also \( 1 \).

Hence, the horizontal asymptote is given by:
\[
y = \frac{1}{1} = 1.
\]

Putting this all together, the asymptotes for the function \( f(x) = \frac{x - 1}{x - 4} \) are:
- Vertical asymptote: \( x = 4 \)
- Horizontal asymptote: \( y = 1 \)

Thus, the correct choice from the options given is:
**Vertical asymptote: x = 4, horizontal asymptote y = 1**.