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?

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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