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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:

Inverse Function

Functions		Inverse Function
1.	f:[0,∞)→[0,∞) defined by f(x)=x²	f^-1:[0,∞)→[0,∞) defined by f^-1(x) = √x
2.	f:[-∏/2,∏/2] →[-1,1] defined by f(x)=sin x	f¹ [-1,1]→[-(∏/2),∏/2] defined by f^-1(x)=sin^-1x
3.	f:[0,∏]→[-1,1] defined by f(x)=sinx	f¹:[-1,1]→[0,∏] defined by f¹(x)=cos^-1x
4.	f:[-∏/2,∏/2] →(-∞,∞) defined by f(x)=tan x	f¹:(-∞,∞)→[-(∏/2),∏/2] defined by f¹(x)=tan^-1 x
5.	f:(0,∏)→(-∞,∞) defined by f(x) = cot x	f^-1:(-∞, ∞)→(0,∏) defined by f^-1(x)=cot^-1 x
6.	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
7.	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
8.	f:R → R⁺ defined by f(x) = e^x	f^-1(x):R⁺ → R defined by f^-1 (x) = In x.

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