Vertical asymptotes are found by setting denomintor factors equal to zero.
x(x + 1)
Since the horizontal asymptote is y=0, the degree of numerator is less than degree of denominator.
x-intercept is the value that makes the numerator zero. Its factor is (x - 1).
The initial value condition allows you to find the leading coefficient.
So here is our function so far.
f(x) = C(x - 1)/ x(x + 1)
1 = C(2 - 1) / [2(2 + 1)]
1 = C / 6
C = 6
Therefore, your function is
f(x) = 6x / (x2 + x)