How to Find the Inverse of a Function? Calculation & Example

Today you will learn "how to find the Inverse of a function". 

Before knowing how to find the inverse of a function, we need to know what is the inverse of a function, after that we will understand their calculation process step by step with an example.

So, without wasting time let's get started.

What is the Inverse of a Function?

If f is a function then the set of ordered pairs obtained by interchanging the first and second coordinates of each ordered pair in f is called the inverse of f. It is denoted by f ⁻¹.

A function f has an inverse if and only if f is injective and f is injective and surjective.

Let f: A → B be a function. The inverse of f is another function f - ¹: B → A.

So,

x = f - ¹( y ) ⇔ y = f ( x )

how to find the Inverse of a function

How to Find the Inverse of a Function?

If the function f is given, then the inverse function f ⁻¹ can be found by an algebraic method that involves the following steps. 

Step - 1 

Write the function y = f ( x ) form . 

Step - 2 

Solve the equation in Step - 1 for x in terms of y. 

Step - 3 

In the resulting equation in Step - 2, replace x with f⁻¹ (y). 

Step - 4 

Replace each y in the result of Step - 3 by x to get f ⁻¹ ( x ). 

Step - 5 

Check the answer by verifying that 

f ⁻¹ ( f ( x ) ) = x .

Example

Let f: R → R be the function defined by 

f ( x ) = 2x - 7. find f ⁻¹ ( x )?

Solution

Given that,

f( x ) = 2x - 7 , 

we are to find f - ¹ ( x ) 

Step - 1 

Write the function y = f(x)

So,

y = 2x - 7

Step - 2 

Now solving the equation for x in terms of y so,

y = 2x - 7

2x = y + 7

Step - 3

Now replace the x with f ⁻¹(y)

So,

f ⁻¹(y) = (y +7)/2

Step - 4

Now replace each y by x to get f ⁻¹(x)

So,

f ⁻¹(x) =  (x +7)/2

Step - 5

Now we will check the answer by verifying that

f ⁻¹ ( f ( x ) ) = x . 

So,

We have f ( x ) = 2x - 7 and f ⁻¹(x) = (x +7)/2

Now,

f ⁻¹{f(x)}  =  f ⁻¹(2x - 7)
 
              =  (2x - 7 + 7)/2

              =   2x/2

             =   x

Hence prove it,

          f ⁻¹ ( f ( x ) ) = x 

Question

If f : R → R be the function defined by 
f ( x ) = (x - 3)/2. find f ⁻¹ ( x ) and also verify?

Solution

Let,

y  = f(x)

So,

y = (x-3)/2

Swap x with y so,

x =  (y -3)/2

2x = y - 3

y = 2x + 3

Since,

y = f ⁻¹(x)

So,

f ⁻¹(x)  = 2x + 3


So here we explained step by step how to find the inverse of a function with an example if you are still confused then you can contact me via comment or via mail.

Thank You.

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