Write a function whose graph has ahole, a vertical asymptote, and a horizontal asymptote

Answer 1

Here is an example of a function, f(x), that has a hole at x = 1, a vertical asymptote at x = 2, and a horizontal asymptote at y = 3:

f(x) = (x-1)(x-2)/(x-2)

The hole at x = 1 is created by cancelling out the (x-1) term in the numerator with the (x-1) term in the denominator.

The vertical asymptote at x = 2 is created by the denominator (x-2) which causes the function to become undefined at x = 2.

The horizontal asymptote at y = 3 is created by the fact that as x approaches infinity, the (x-1) term in the numerator becomes insignificant compared to the (x-2) in the denominator, so the function tends towards y = 3.

The graph of this function would have a hole at x = 1, a vertical asymptote at x = 2, and a horizontal asymptote at y = 3.