How to make InverseFunction work on function defined with restriction on arguments

Question

This example works to define InverseFunction on the first argument of a function

f = Function[{##} /. {x_, y_} :> Beta[x, y] - x + y];
g = InverseFunction[f, 1, 2];
g[3, 4]
(*Root[{1 + Beta[#1, 4] - #1 &, 1.17659010788279817113535497590}]*)
N[%]
(* 1.17659 *)

When changing the definition of f to use x_Integer instead of x it no longer works

f = Function[{##} /. {x_Integer, y_Integer} :> Beta[x, y] - x + y];
g = InverseFunction[f, 1, 2];
g[3, 4]

Returns unevaluated

InverseFunction[{##1} /. {x_Integer, y_Integer} :> Beta[x, y] - x + y &, 1, 2][3, 4]

How to make InverseFunction work on functions defined with restrictions on its arguments? I looked at ConditionalExpression but could not figure how to use it here.

V 9.01 on windows

Answer 1

Since for real input there isn't any errors spat out, I assume the function is invertible in the first variable so one way to get the function you want is to constrain the inverse to be an integer. A way to do this is:

Clear[f, g, intg]
f[x_, y_] := Beta[x, y] - x + y;
g[z_, y_] := InverseFunction[f, 1, 2][z, y];
intg[z_, y_Integer] :=
 If[IntegerQ@# || IntegerPart@# == #, IntegerPart@#,] &@g[z, y]

The second condition in If is so that if the inversion is done numerically you force the result to be an integer (but this is to some accuracy i.e. 4 == 4. (1 + 1 10^-14) is True while 4 == 4. (1 + 1 10^-13) is False).

Now

f[3, 2]
(* -11/12 *)

and

intg[-11/12,2]
(* 3 *)

but

intg[-0.91666,2]

is unevaluated.