# Inverse Function.

**Inverse of a Function:**

Let “f” be a Function from set A to B or , elements of set A are changed or outputted to elements of set B if they are processed through or inputed in the function “f”.

or f:A→B

Then , inverse of the function “f” is a new rule or function (Let the new function be function “g”) , such that if we input the output of function “f” into the new function “g” then it produces or outputs the input of function “f”

or,

inverse of a function “f” f:A→B is another function”g” such that g:B→A

**Denotation of Inverse Function:**

As we know , A function(f) from set A to set B is denoted by:

f:Ar→B

the inverse of the function “f” is denoted by adding a “-1″ superscription in the function and interchanging the position of input and output set of the function. Like inverse of function “f” denoted above is denoted as:

f^{-1}:B→A

We can denote the inverse of a function diagrammatically as following:

And , as we denoted the process of processing of input (x) by a function (f) and producing output (y) by:

f(x)=y

We denote the same fact for inverse of the function (f) by:

f^{-1}(y)=x

which means the inverse function “f^{-1}” processes “y” which is output of function “f” and outputs “x” which is input of function “f”.

** Note: **

f^{-1}≠1/f

**There is no Inverse of Onto Function:**

A function “f’ can only have a defined inverse function “f^{-1}“only if it is a One-To-One function. A Onto Function can never have a inverse practically.

For Example:

If A={1,2,3} and B={a,b} and function “f” is defined by:

Then , we cannot make a inverse of function “f” because inverse of function “f” should map “a” to “1″ and “b” to “2″ and “3″ both.

But practically we can never make a rule or function which have two or variable output of same input.

**Some Examples of Inverse Function:**

a> Let f:A→B be a function which is defined by the arrow diagram on the top of the image below. Then the inverse function of “f” or , f^{-1}:B→A is defined by the arrow diagram on the bottom of the image below.

b> If f:A→B is a function defined by f(x)=x^{2} then , the inverse function of “f” is “f^{-1}” which is defined by:

f(x^{2})=x

or, f(x)=√x

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