(1) HA AT y=⅓
The degrees of the numerator and denominator must be the same and the coefficient of the highest degree in the denominator must be 3 times that in the numerator.
(2) VA AT x=-4/3
The factor x+4/3 or 3x+4 must appear in the denominator. 3x+4 would also satisfy (1) in the absence of any other coefficients in the numerator.
(3) x-INTERCEPT (-1,0)
Numerator must contain the factor x+1.
(4) y-INTERCEPT (0,¾)
The factor x+3n (n≠0 is an integer) must appear in the numerator and, because of (2) x+n must be a factor in the denominator. This assumes there are no other constants in the denominator factors apart from 3x+4. If there are other constants, and their product is p then x+n/p replaces x+n. In particular, when p=-1, this changes x+n to x-n.
(5) HOLE AT (½,³⁄₁₁)
The factor x-½ or 2x-1 must appear in both numerator and denominator and when these are removed, the numerical value of the expression must be ³⁄₁₁ when x=½. When constructing the function the common factor can be omitted until the last stage.
y=(x+1)(x+3n)...(2x-1)/((3x+4)(x+n)...(2x-1)).
For all values of x≠½, (2x-1)/(2x-1)=1.
When x has a large magnitude, y≈xⁿ/3xⁿ=⅓;
When x=0, y=3n/(4n)=¾.
When x=-⁴⁄₃, there is a VA.
When x=½, y=³⁄₁₁:
3/11=(½+1)(½+3n).../((³⁄₂+4)(½+n)...,
3/11=3(1+6n).../(11(1+2n)...
For this to happen, (1+6n).../((1+2n)...)=1 when x=½ and n≠0.
But any other factors will produce a different y-intercept (see 4), so it will not be possible to produce a rational function satisfying all the given conditions.