Question

Find all the asymptotes for f(x) = (x-4)/(x-9).

a) Vertical asymptote at x = 9
b) Horizontal asymptote at y = 1
c) No asymptotes
d) Oblique asymptote at y = x - 5

Answer 1

The function has a vertical asymptote at x = 9 and no other asymptotes.

Step-by-step explanation:

This function has a vertical asymptote at x = 9 because as x approaches 9, the denominator approaches zero, and the function approaches infinity. We can verify this by examining the function and noting that the denominator becomes zero when x is 9, which is not allowed because division by zero is undefined. The given function is f(x) = (x-4)/(x-9).

To find the asymptotes, we can analyze the behavior of the function as x approaches certain values.

a) Vertical asymptote at x = 9:

When x approaches 9 from the left side, f(x) approaches negative infinity. When x approaches 9 from the right side, f(x) approaches positive infinity.

b) No horizontal asymptote:

As the degree of the numerator and denominator is the same, there is no horizontal asymptote.

c) No asymptotes:

Since we only have a vertical asymptote, there are no other asymptotes.

d) No oblique asymptote:

Since there is no horizontal asymptote, there is no oblique asymptote.

Answer 2

The function f(x) = (x-4)/(x-9) has a vertical asymptote at x = 9 and a horizontal asymptote at y = 1. There are no oblique asymptotes because the degrees of the numerator and the denominator are equal.

Step-by-step explanation:

To find all the asymptotes for the function f(x) = (x-4)/(x-9), we need to look at places where the function is undefined and its behavior at infinity.

First, the vertical asymptote occurs where the denominator is zero since the function cannot have a value at that point. Setting the denominator equal to zero gives us x - 9 = 0, thus the vertical asymptote is at x = 9.

Next, we look for a horizontal asymptote by determining the limit of f(x) as x approaches infinity. The degrees of the numerator and denominator are the same (both are 1), so the horizontal asymptote is the ratio of the leading coefficients. This gives us a horizontal asymptote at y = 1.

Finally, since the degrees of the numerator and denominator are the same, there is no oblique asymptote. Oblique asymptotes only occur if the degree of the numerator is exactly one more than the degree of the denominator.

Therefore, the correct answers are:
a) Vertical asymptote at x = 9
b) Horizontal asymptote at y = 1
c) There are asymptotes, so this option is incorrect
d) There is no oblique asymptote, so this option is incorrect