Describe the horizontal and vertical asymptotes of f(x)=log4 x

Question

.(1 point)
Responses

The function f(x) has a vertical asymptote at x = 0 and no horizontal asymptote.
The function f(x) has a vertical asymptote at x = 0 and no horizontal asymptote.

The function f(x) has a horizontal asymptote at y = 0 and a vertical asymptote at x = 4.
The function f(x) has a horizontal asymptote at y = 0 and a vertical asymptote at x = 4.

The function f(x) has a horizontal asymptote at y = 4 and a vertical asymptote at x = 0.
The function f(x) has a horizontal asymptote at y = 4 and a vertical asymptote at x = 0.

The function f(x) has a horizontal asymptote at y = 0 and no vertical asymptote.

Answer 1

For the function \( f(x) = \log_4 x \):

1. Vertical Asymptote: The logarithmic function \( \log_4 x \) is undefined for \( x \leq 0 \) and approaches negative infinity as \( x \) approaches 0 from the right. Therefore, there is a vertical asymptote at \( x = 0 \).

2. Horizontal Asymptote: As \( x \) approaches infinity, \( f(x) \) continues to increase without bound; thus, there is no horizontal asymptote.

Given these observations, the correct response would be:

The function f(x) has a vertical asymptote at x = 0 and no horizontal asymptote.