Question

Can the graph of y = f(x) intersect a horizontal asymptote?

1) Yes
2) No
3) It depends on the function
4) Cannot be determined

Answer 1

Yes, the graph of y = f(x) can intersect a horizontal asymptote. It depends on the function and how it is defined.

Step-by-step explanation:

Yes, the graph of y = f(x) can intersect a horizontal asymptote. It depends on the function and how it is defined. An asymptote is a line that a graph approaches but never touches or crosses. However, there are some functions, such as rational functions, that can intersect a horizontal asymptote. For example, the graph of y = 1/x has a horizontal asymptote at y = 0, but it intersects the asymptote at x = 1.

Answer 2

The graph of y = f(x) can intersect a horizontal asymptote depending on the function.

The answer is option ⇒3) It depends on the function

Step-by-step explanation:

A horizontal asymptote is a horizontal line that the graph of a function approaches as x approaches positive or negative infinity. It represents a limit to which the function's values tend to as x becomes very large or very small.

There are three possible scenarios regarding the intersection of the graph of y = f(x) with a horizontal asymptote:

1) The graph does not intersect the horizontal asymptote: In this case, the function approaches the asymptote but never crosses it. The graph may get arbitrarily close to the asymptote, but it never actually touches or crosses it.

2) The graph intersects the horizontal asymptote at a single point: In some cases, the graph of y = f(x) may cross the horizontal asymptote at a single point and then continue its path. This occurs when the function has a point where the value of y equals the value of the horizontal asymptote.

3) The graph intersects the horizontal asymptote multiple times: It is also possible for the graph of y = f(x) to intersect the horizontal asymptote multiple times. This happens when the function has multiple points where the value of y equals the value of the horizontal asymptote.

The specific behavior of a function in relation to a horizontal asymptote depends on the nature of the function itself. Different functions can exhibit different intersection patterns or behaviors with respect to a horizontal asymptote.

The answer is option ⇒3) It depends on the function