Question

whats the domain and range of y=f -1 (x), in inequality form?

Comments

We need a definition of f(x) to answer this question.

Answer 1

If y=f^-1(x) (as I read the question), then x=f(y).

Let's use some simple functions for f.

(1) x=e^y, then y=ln(x) and the domain is x>0 and the range is -∞≤y≤∞ (unbounded).

(2) x=sin(y), then y=sin^-1(x) and the domain is -1≤x≤1, in other words the domain is between -1 and 1, while the range is unbounded, that is, is between -∞ and +∞.

(3) x=y² which is a sideways parabola. (a) y=√x or (b) y=-√x are two possible inverses, so the ranges are the same but the domains are different. The domains are both x≥0 but (a) has the range y≥0 while (b) has the domain y≤0. 

(4) x=1/y, then y=1/x (rectangular hyperbolae) and the range and domain are x≠0 and y≠0 respectively, that is, unbounded except for x=0.