A rational function is one that has a numerator and a denominator.
Vertical asymptotes are values of x in which the function gets closer to, but never touches. We can find the vertical asymptotes by setting the denominator equal to zero.
Since the vertical asymptote is x=-2 and x=0, our denominator of the function will be
x(x + 2) = x2 + 2x
Horizontal asymptotes are values of y in which the function gets closer to, but never touches. Since the horizontal asymptote is y=0 , the degree of the numerator must be less than the degree of the denominator. And since the x-intercept is x=1, our numerator will be
x - 1
So far, we can assume the function to be
f(x) = (x - 1) / (x2 + 2x)
Next, we need to use the condition f(2)=1. If we plug in x=2, f(x) should equal 1. But we need to multiply the right side of the function by some constant C.
1 = C(2 - 1) / (22 + 2(2))
Solve for C.
1 = C / 8
8 = C
So the function will be
f(x) = [8(x - 1)] / [(x2 + 2x)]
f(x) = (8x - 8) / (x2 + 2x)