Answer:
This function has a horizontal asymptote at y=0, a vertical asymptote at x=-1, and a hole at (1,2).
Explanation:
A rational equation with the given requirements can be written in the form:
f(x) = (x - 1) / [(x + 1)g(x)]
where g(x) is a factor in the denominator that ensures the vertical asymptote at x=-1.
To meet the condition that y=0 is a horizontal asymptote, we need to ensure that the degree of the denominator is greater than or equal to the degree of the numerator.
To create a hole at (1,2), we need to ensure that the factor (x-1) appears in both the numerator and the denominator, so that they cancel each other out at x=1.
One possible function that meets all of these requirements is:
f(x) = (x - 1) / [(x + 1)(x - 1)]
Simplifying this function, we get:
f(x) = 1 / (x + 1)
This function has a horizontal asymptote at y=0, a vertical asymptote at x=-1, and a hole at (1,2).