INVERSE FUNCTION

Inverse Function : 

Let f be a one-one onto function from A to B.

Let y be an arbitrary element of B.

Then f being onto, there exists an element x in A such that 

f(x)  =  y

So, we may define a function, denoted as f^-1 as

f^-1 : B ---> A : f^-1(y)  =  x

if and only if f(x)  =  y.

The above function f^-1 is called the inverse of f.

A function is invertible, if and only if f is one-one onto. 

More clearly, one-one onto function will have an inverse function

You may be able to find f^-1 for any function f.

But, f^-1 will also be a function, only if f is one-one onto function.  

Remarks : 

If f is one-one onto, then f^-1 is also one-one onto. 

Illustration : 

If f : A ---> B, then

f^-1 : B---> A

Finding Inverse of a Function - Steps

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

Step 1 :

In the above function, f(x) has 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".

Therefore, inverse of f(x) is, 

f^-1(x)  =  x - k 

Finding Inverse of a Function - Examples

Problem 1 :

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

Subtract 3 from each side. 

y - 3  =  2x

Divide each side by 2. 

(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".

So, inverse of f(x) is, 

f^-1 (x)  =  (x - 3)/2

Problem 2 :

Find the inverse of the function h(x)  =  log₁₀(x). 

Solution :

Step 1 :

Given function : h(x)  =  log₁₀(x)

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

Then, we will get 

y  =  log₁₀(x)

y  =  log₁₀(x) has been defined by "y" in terms of "x".

Step 2 :

Now we have to redefine y  =  log₁₀(x) by "x" in terms of "y". 

y  =  log₁₀(x)

10^y  =  x

x  =  10^y

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

Step 3 :

In x  =  10^y replace "x" by h^-1(x) and "y" by "x". 

So, inverse of h(x) is,

h^-1(x)  =  10^x

After having gone through the stuff given above, we hope that the students would have understood, how to find inverse of a function. 

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