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.

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

Solution: We have f(f-1(x)) = x

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

        =>  e2f-1(x) - 2xef-1(x)  -1 = 0

        =>  ef-1(x)  = x + √(x2 +1).

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

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

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

Inverse Function


Inverse Function


f:[0,∞)→[0,∞) defined by f(x)=x2

f-1:[0,∞)→[0,∞) defined by f-1(x) = √x


f:[-∏/2,∏/2] →[-1,1] defined by f(x)=sin x

f1 [-1,1]→[-(∏/2),∏/2]  defined by f-1(x)=sin-1x


f:[0,∏]→[-1,1] defined by f(x)=sinx

f1:[-1,1]→[0,∏] defined by f1(x)=cos-1x


f:[-∏/2,∏/2] →(-∞,∞) defined by f(x)=tan x

f1:(-∞,∞)→[-(∏/2),∏/2] defined by f1(x)=tan-1 x


f:(0,∏)→(-∞,∞) defined by f(x) = cot x

f-1:(-∞, ∞)→(0,∏) defined by f-1(x)=cot-1 x


f:[0,∏/2)U(n/2,n]→(-∞, -1]U[1,∞) defined by f(x) = sec x

f-1:(-∞,-1]U[1,∞) →[0,∏/2)U(∏/2,∏]  defined by f-1 (x) = sec-1 x


f:[-(∏/2),0)(0,n/2]→(-∞,-1]U[1,∞) defined by f(x) = cosec x

f-1:(-∞,-1]U[1,∞) →[0,-(∏/2))U(0,∏/2]  defined by f-1 (x) = cosec-1 x


f:R → R+ defined by f(x) = ex

f-1(x):R+ → R defined by f-1 (x) = In x.

A comprehensive study material for IIT JEE, JEE Main /Advanced and other engineering examinations is available online free of cost at Study Set Theory, Functions and a number of topics of Algebra at askIITians website. The website has links to numerous live online courses for IIT JEE preparation - you do not need to travel anywhere any longer - just sit at your home and study for IIT JEE live online with

To read more, Buy study materials of Set Relations and Functions comprising study notes, revision notes, video lectures, previous year solved questions etc. Also browse for more study materials on Mathematics here.

Course Features

  • Video Lectures
  • Revision Notes
  • Previous Year Papers
  • Mind Map
  • Study Planner
  • NCERT Solutions
  • Discussion Forum
  • Test paper with Video Solution