f[x_]:=1/(x-3)

3

f[x_]:=1/(x-3);
FunctionDomain[f[x], x, Reals]
(* x<3||x>3 *)
Not[%]//FullSimplify
(* x == 3 *)

Not[FunctionDomain[f[x], x,Reals]]//FullSimplify
(* x == 3 *)

Not[FunctionDomain[f[x], x,Reals]]//FullSimplify //Last
(* 3 *)

Not[FunctionDomain[Tan[x], x,Reals]]//FullSimplify
(* 1/2 + x/π ∈ Integers *)

Not[FunctionDomain[(x + y)/(x^2 - y^2), {x, y},Reals]]//FullSimplify
(* x^2==y^2 *)

Not[Reduce[Abs@f[x] < Infinity, x, Reals]] // FullSimplify // Last
(* 3 *)

f[x_] = Product[1/(x - a[i]), {i, 3}]

Solve[1/f[x] == 0, x]

f[x_, y_] = Product[1/(x + y - a[i]), {i, 3}]
Solve[1/f[x, y] == 0, x]