A special function

A function f: X → Y has an inverse iff it has a one-to-one mapping from the set X to Y.

If we're transforming a line, f(x)=1, we could reparametrize this line in vector form as [x,f(x)] = [t, 0]. Then if we subject this to a one-to-one matrix transformation A[x,f(x)]+B, where A and B are matrices with constant entries, we can arrive at [x,f(x)]=[0,t]. Even though x → f(x) is no longer one-to-one, the resulting matrix is one-to-one. :|

TL;DR: In order to have an invertible function:

• f must be an injective function - If f is continuous, then f must either be strictly increasing or decreasing in order to be injective.
  
• f(f⁻¹(x)) = f⁻¹(f(x)) = x, not their limits
  

What about periodic functions, like sin(x) and cos(x)? Are asin(x) and acos(x) somehow not true inverses in comparison to arcsin(x) and arccos(x)? (By asin(x) and acos(x), I mean the programming command that spans the entire range, unlike arcsin(x) and arccos(x), which restricts the domain.)

Hmmm so there's no simple function that works like that? (Read: defined in Reals)

 Spoiler

1. f(x)=0 and f⁻¹(x)=∞, or x=0
2. f(x)=∞, or x=0, and f⁻¹(x)=0

Alternatively, you could force f(x) = x^±i to return real numbers by having your domain be a set of complex numbers.