# Inverse Functions

## Introduction

The concept of function is of paramount importance in mathematics and among other disciplines as well. An inverse function is a function that undoes the action of another function. Let us recapitulate what we know about functions that are relevant to the understanding of inverse functions.

### Function as a set of ordered pairs

Let A and B be two non-empty sets. A relation from A to B i..e a subset of A x B is called a function or a mapping or a map from A to B if,

1. For each a ∈ b, there exists b ∈ B such that ( a, b ) ∈ f
2. ( a, b ) ∈ f and ( a, c ) ∈ f ⇒ b = c

Thus a non-void subset of A x B is a function from A to B if each element of A appears in some ordered pair in f and no two pairs in f have the same first element.

### Domain, Co-domain and Range of a Function

Let f : A → B. then the set A is known as the domain of f and the set B is known as the range co-domain of f. the set of all f-images of elements of A is known as the range of f or image set of A under f and is denoted by f ( A ).

Thus f ( A ) = { f (x)  : x ∈ A } = Range of f

### Onto Functions

A function A → B is said to be onto function or a surjection if every element of B is the f-image of some element of A, i.e. f ( A) = B or range of f is the c-domain of f.

Thus f : A → B is an onto function iff for each b ∈ B there exists a ∈ A such that f ( a ) = b

### One-One Functions

A function A → B is said to be a one-one function or an injection if different elements of A have different images of B.

Thus f : A → B is one-one

⇔ a ≠ b ⇒ f ( a ) ≠  f ( b ) for all a, b ∈ A

⇔ f ( a )  = f ( b ) ⇒ a = b for all a, b ∈ A

### Many-One Functions

A function A → B is said to be a many-one function if two or more elements of set A have the same image in B.

Thus f : A → B is many-one if there exists x, y ∈ A such that x≠ y but f ( x ) = f ( y )

Now let us understand what we mean by inverse functions.

## What is an Inverse Function?

Let A and B be two sets and let f :  A → B be a function. If we follow a rule in which elements of B are associated to their pre-images, then we will find that under such a rule there may be some elements in B which are not associated to elements in A. this happens when f is not an onto map. Therefore all elements in B will be associated to elements in A if f is an onto map. Also, if it is a many-one function then under the said rule an element in B may be associated to more than one element in A. Therefore, an element in B will be associated to a unique element in A if f is an injective map.

We can infer from the above discussion that if f : A → B is a bijection, we can define a new function from B to A which associates each element y ∈ B to its pre-image f –1 ( y ) ∈ A. Such a function is known as the inverse of function f and is denoted by f -1 . Therefore we can now define an inverse function as:

Let f : A → B be a bijection. Then a function g : B → A which associates each element y B to a unique element x A such that f ( x )  = y is called the inverse of f. This means,

f ( x ) = y g ( y )  = x

Therefore, if f : A → B is a bijection function, then f-1 : B → A such that f ( x ) = y ⇔ f -1 ( y )  = x

Functions that have inverse are called one-to-one functions.

## Algorithm to Find the Inverse of a Function

Now that we have understood what we mean by an inverse function it is important to learn how to actually find the inverse of a given function. The following algorithm defines the steps that are to be followed in order to find the inverse of a given function –

Put f ( x)  = y where y ̬∈ B and x ∈ A

Now, solve f ( x ) = y to obtain x in terms of y.

In the relation obtained in the above step, replace x by f -1 ( y ) to obtain the required inverse of f.

Let us understand this through an example.

Example

Find the inverse for the function f ( x ) = 3x−4. State the domain and range of the inverse function. Verify that f−1( f ( x ) ) = x

Solution

We will find the inverse of the function through the following steps –

First, we have been given that f ( x ) = 3x – 4.

Therefore, if  y = 3x – 4, this means that 3x = 4 + y which further implies that –

$x=\frac{1}{3}y+\frac{4}{3}$

Now, we will rewrite the above equation as:

 and let y be equal to f-1 ( x )

Hence, we will have,

f-1 ( x ) =$y=\frac{1}{3}x+\frac{4}{3}$

No, since the domain of f is (−∞,∞) and the range of f−1 is (−∞,∞) and since the range of f is (−∞,∞) and the domain of f−1 is (−∞,∞).

Therefore,

Domain of f−1 is (−∞,∞).

Range of f−1 is (−∞,∞)

Now, let us move to the second part of the question.

We are required to verify that f−1(f(x) = x

In order to verify this, we will write the function as

f−1 (f(x)) = f−1 (3x−4) = $\frac{1}{3}(3x-4)+\frac{4}{3}$

$=x-\frac{4}{3}+\frac{4}{3}=y$

Hence f−1( f ( x ) ) = x

It is important to note here that for f -1(f(x)) to be the inverse of f ( x) both  f 1(f(x)) = x and f ( f-1 (x)) = x for all x in the domain of the inside function.

Example

Let f : N → R be a function defined as f ( x ) = 4x2 + 12x  + 15. Show that f : N → Range ( f ) is invertible. Find the inverse of f.

Solution

We have been given that f : N → R be a function defined as f ( x ) = 4x2 + 12x  + 15. We are required to prove that f : N → Range ( f ) is invertible and then find the inverse of f.

In order to prove that f is invertible, is tis sufficient for us to show that f : N → Range ( f ) is a bijection.

f is one-one. For any x, y ∈ N, we find that,

f ( x ) = f ( y )

⇒ 4x2 + 12x  + 15 = 4y2 + 12y + 15

⇒ 4 (x2 – y2 ) + 12 ( x – y ) = 0

⇒ ( x – y ) ( 4x  + 4y + 3 ) = 0

⇒ x – y = 0

⇒ x = y

Therefore, f : N → Range ( f ) is one-one.

Obviously, f : N → Range ( f ) is onto. Hence, f : N → Range ( f ) is invertible.

Now, let us find the inverse of the function.

Let f-1 denote the inverse of f, then

fof -1 ( x ) = x for all x ∈ Range ( f )

⇒ f ( f -1 ( x )) = x for all  x ∈ Range ( f )

⇒ 4 f -1 ( x )2 + 12 f -1 ( x ) + 15 = 0 for all x ∈ Range ( f )

4 { f -1 ( x )2 + 12 f -1 ( x ) + 15 – x  = 0

Solving this equation for f-1 ( x ), we get

f-1 ( x ) = $\frac{-12\pm \sqrt{(144-16 (15-x ))}}{8}$

⇒ f-1 ( x ) = $\frac{-12\pm \sqrt{(16x-96)}}{8}=\frac{-3\pm \sqrt{(x-6)}}{2}$

⇒ f-1 ( x ) = $\frac{-3\pm \sqrt{(x-6)}}{2}$

Hence, the inverse of f = $\frac{-3\pm \sqrt{(x-6)}}{2}$

## Properties of an Inverse Function

The following are the properties of an inverse function.

1. The inverse function of a bijection is unique.
2. The inverse of a bijection is also a bijection.
3. If f : A → B is a bijection and g : B → A is the inverse of f then fog = IB and gof = IA, where IA and IB are the identity functions on the sets A and B respectively.
4. If f : A → B and g : B → C are two bijections, then gof : A → C is a  bijection and (gof)-1 = f-1og-1
5. Let f : A → B and g : B → C be two functions such that gof = IA, and fog = IB. then f and g are bijections and g = f-1.
6. Let f : A → B be an invertible function. Then the inverse of f-1 is f.

## Relation between Graphs of a function and Its Inverse

The graph of a bijection f and its inverse f-1 are closely related.  If ( a, b ) is a point on the graph of f, then b = f ( a ) and a =  f -1 ( b ). As b is the domain of f-1, therefore, ( b, f – 1 ( b ) ) is the point on the graph of f-1. But ( b, f – 1 ( b ) ) = ( b, a ). Therefore, ( b, a ) is on the graph of f-1.

Thus if ( a, b ) is a point on the graph of f, then ( b, a ) is a point on the graph of f-1. But, ( a, b ) and (b, a  ) are reflections of one another in the line y =  x. Thus the graph of f-1 may be obtained by reflecting the graph of f in the line mirror y = x.  this means that graphs of f and f-1 are mirror images of each other in the line mirror y =  x.

From the above discussion, we can also infer that if the graphs of f ( x ) and f -1( x ) intersect each other, their points of intersection lie on the line y = x. consequently, solutions of the equation f ( x ) = f -1 (x) are same as that of f ( x ) = x or f-1 ( x ) = x.

Do we have an inverse function of every function? Let us find out.

## Determining the inverse of a function

It is important to note about the inverse function is that the inverse of a function is not the same as its reciprocal, i.e., f – 1 (x) ≠ 1/ f(x). Therefore, not all functions have an inverse, and hence it is important to check whether a function has an inverse before embarking on determining its inverse. So how do we check whether an inverse function exists for a function or not? The following rule applies in this case –

A function f has an inverse if and only if when its graph is reflected about the line y = x, the result is the graph of a function. The following test is used to determine whether the function has an inverse or not.

### Horizontal Line Test

The following steps are followed in the horizontal line test to determine whether there exists the inverse of a function.

1. Let f be a function.
2. If any horizontal line intersects the graph of f more than once, then f does not have an inverse.
3. If no horizontal line intersects the graph of f more than once, then f does have an inverse.

Let us understand this by an example

Example

Graph the function, f(x) =  (x – 3 ) x2 and determine whether or not it has an inverse.

Solution

The graph of the given function will be

We can see that there is a horizontal line that hits the graph more than once.

This means that the function f is not one-to-one.

Hence, we can say that the function f does not have an inverse.

## Remember

1. A non-void subset of A x B is a function from A to B if each element of A appears in some ordered pair in f and no two pairs in f have the same first element.
2. Let f : A → B. then the set A is known as the domain of f and the set B is known as the range co-domain of f.
3. Let f : A → B be a bijection. Then a function g : B → A which associates each element y ∈ B to a unique element x ∈ A such that f ( x )  = y is called the inverse of f.
4. A function A → B is said to be onto function or a surjection if every element of B is the f-image of some element of A, i.e. f ( A) = B or range of f is the c-domain of f.
5. A function A → B is said to be a one-one function or an injection if different elements of A have different images of B.
6. A function A → B is said to be a many-one function if two or more elements of set A have the same image in B.
7. The inverse function of a bijection is unique.
8. The inverse of a bijection is also a bijection.
9. If f : A → B is a bijection and g : B → A is the inverse of f then fog = IB and gof = IA, where IA and IB are the identity functions on the sets A and B respectively.
10. If f : A → B and g : B → C are two bijections, then gof : A → C is a  bijection and (gof)-1 = f-1og-1
11. Let f : A → B and g : B → C be two functions such that gof = IA, and fog = IB. then f and g are bijections and g = f-1.
12. Let f : A → B be an invertible function. Then the inverse of f-1 is f.
13. The graph of f-1 may be obtained by reflecting the graph of f in the line mirror y = x.
14. A function f has an inverse if and only if when its graph is reflected about the line y = x, the result is the graph of a function.