Inverse Function

Let f : X → Y be a function defined by y = f(x) such that f is both one - one and onto. Then there exists a unique function g : Y → X such that for each y ε Y,

g(y) = x <=> y =  f(x). The function g so defined is called the inverse of f.

Further, if g is the inverse of f, then f is the inverse of g and the two functions f and g are said to be the inverses of each other. For the inverse of a function to exists, the function must be on-one and onto.

Method to Find Inverse of a Function

If f^-1 be the inverse of f, then fof^-1 = f^-1 of = I, where I is an identity function.

fof^-1 = I => (fof^-1(x)) = I (x) = x.

Apply the formula of f on f^-1 (x), we will get an equation in f^-1 (x) and x.

Solve it to get f^-1 (x).

Note : A function and its inverse are always symmetric with respect to the line y = x.

Illustration:     
Let f : R → R defined by f(x) = (e^x-e^-x)/2 . Find f^-1 (x).

Solution: We have f(f^-1(x)) = x

        =>  (e^f-1(x) - e^-f-1(x))/2 = x

        =>  e^2f-1(x) - 2xe^f-1(x)  -1 = 0

        =>  e^f-1(x)  = x + √(x² +1).

But negative sign is not possible because L.H.S. is always positive.

Thus e^f-1(x) = x + √(x² +1) . Hence, f^-1(x) = log(x + √(x² +1)) .

We give below some standard functions along with their inverse functions: