Use Coupon: CART10 and get 10% off on all online Study Material

Total Price: R

There are no items in this cart.
Continue Shopping
Get instant 50% OFF on Online Material.
Use coupon code: MOB50 | View Course list

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




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, AIEEE 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 know more about the study material of engineering and medical exams, please fill up the form given below:

We promise that your information will be our little secret. To know more please see our Privacy Policy
We promise that your information will be our little secret. To know more please see our Privacy Policy


Signing up with Facebook allows you to connect with friends and classmates already using askIItians. It’s an easier way as well. “Relax, we won’t flood your facebook news feed!”