# THE INVERSE FUNCTION AND ITS PROPERTIES

## About "The inverse function and its properties"

The inverse function and its properties :

In mathematics, an inverse function is the one which  "reverses" another function.

For example, in a function "f", if we give input "x", we will get the output "y".

Let "g" be the inverse function of "f".

In "g", "y" will be the input and "x" will be the output.

That is, f(x)  = y if and only if g(y)  =  x

Or g(x)  =  f¹(x)

## How to find inverse of a function ?

We have to apply the following steps to find inverse of a function, say f(x).

Step 1 :

Let f(x) = x + k  ("k" is a constant).

In the above function f(x) to be replaced by "y"

Then, we will get  y = x + k.

y = x + k has been defined by "y" in terms of "x"

Step 2 :

Now we have to redefine y = x + k by "x" in terms of "y"

Then we will get x = y - k

Step 3 :

In x = y - k, replace "x" by f⁻¹(x) and "y" by "x".

Hence inverse of f(x) is,  f⁻¹(x) = x - k

## Finding inverse of a function - Example

Question :

Find the inverse of the function f(x) = 2x + 3

Solution :

Step 1 :

Given function : f(x) = 2x + 3

In the above function f(x) to be replaced by "y"

Then, we will get  y = 2x + 3.

y = 2x + 3 has been defined by "y" in terms of "x"

Step 2 :

Now we have to redefine y = 2x + 3 by "x" in terms of "y"

y = 2x + 3 ===> y - 3 = 2x

===> (y-3)/2 = x

===> x = (y-3)/2

Now, the function has been defined by "x" in terms of "y"

Step 3 :

In x = (y - 3)/2, replace "x" by f⁻¹(x) and "y" by "x".

Hence inverse of f(x) is,  f⁻¹(x) = (x - 3)/2

## Properties of inverse function

1.  A function, say "f" is invertible if and only if "f" is one to one onto.

For example, if "f" and "g" are two functions inverse to each other, then both "f" and "g" are one to one onto functions.

2. If f(x) and g(x) are the two functions which are inverse to each other, then we have

f⁻¹(x)  =  g(x) and also g⁻¹(x)  =  f(x)

3.  f(x) and g(x) are the two functions which are inverse to each other, if and only if

fog  =  gof  =  x

4.  Let f(x) be a function such that f : A --> B and "g" be the inverse of "f". Then we have

g : B --> A

5.  Let "f" and "g" be the two functions which are inverse to each other. Then we have

Domain of f(x)  =  Range of g(x)

Range of f(x)  =  Domain of g(x)

After having gone through the stuff given above, we hope that the students would have understood "The inverse function and its properties".

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