Definition


A function  f : A → B is said to be invertible if f is bijective and their inverse function   f-1 : B → A is defined by

 f-1(y) = x if f(x) = y , x ∈ A, y ∈  B

In general, let A be the domain of definition of an injective function f and B the set of its values. By definition, from  y = f-1(B), x = A , it follows that , x = f-1(y), y = Beiθ  , if  x1 ≠ x2 , then  y1 ≠ ythen and conversely [here  y1 =  f(x1),y2  = f(x2)]. It is evident that the domain of definition of f-1 is the set of values of f, and the set of values of f-1 is the domain of definition of the function f .

Also we have

  (f-1 of)(x) = f-1 (f(x)) = f-1(y)  = x   for all x ∈ A 

and (fof-1 )y = f(f-1(y)) = f(x) = y   for all y ∈ B 

Remark

We know that trigonometric functions are periodic functions, and hence, in general, all trigonometric functions are not bijections. Consequently, their inverse do not exist. However, if we restrict their domain and co-domains, they can be made bijections and we can obtain their inverse. The graphs of trigonometric functions gave a fare idea of the domain where they are invertible.