One to one function basically denotes the mapping of two sets. A function g is one-to-one if every element of the range of g corresponds to exactly one element of the domain of g. One-to-one is also written as 1-1. A function f() is a method, which relates elements/values of one variable to the elements/values of another variable, in such a way that elements of the second variable is identically determined by the elements of the first variable.

Apart from the one-to-one function, there are other sets of functions which denotes the relation between sets, elements or identities. They are:

• Many to One function or Surjective function
• Onto Function or Bijective function

## Definition of One-to-One Functions

A function has many types and one of the most common functions used is the one-to-one function or injective function. Also, we will be learning here the inverse of this function.

One-to-One functions define that each element of one set say Set (A) is mapped with a unique element of another set, say, Set (B).

Or

It could be defined as each element of Set A has a unique element on Set B.

In brief, let us consider ‘f’ is a function whose domain is set A. The function is said to be injective if for all x and y in A,

Whenever f(x)=f(y), then x=y

And equivalently, if x ≠ y, then f(x) ≠ f(y)

Formally, it is stated as, if f(x)=f(y)  implies x=y, then f is one-to-one mapped or f is 1-1.

Similarly, if f is a function which is one to one, with domain A and range B, then the inverse of function f is given by;

f-1(y) = x ; if and only if f(x) = y

A function f : X → Y is said to be one to one (or injective function), if the images of distinct elements of X under f are distinct, i.e., for every x1 , x2 ∈ X, f(x1 ) = f(x2 ) implies x1  = x2 . Otherwise it is called many to one function.

In Maths, an injective function or injection or one-one function is a function that comprises individuality that never maps discrete elements of its domain to the equivalent element of its codomain. We can say, every element of the codomain is the image of only one element of its domain.

Examples of Injective Function

• The identity function X → X, is always injective.
• If function f: R→ R, then f(x) = 2x is injective.
• If function f: R→ R, then f(x) = 2x+1 is injective.
• If function f: R→ R, then f(x) = x2 is not an injective function, because here if x = -1, then f(-1) = 1 = f(1). Hence, the element of codomain is not discrete here.
• If function f: R→ R, then f(x) = x/2 is injective.
• If function f: R→ R, then f(x) = x3 is injective.
• If function f: R→ R, then f(x) = 4x+5 is injective.

### Horizontal Line Test

An injective function can be determined by the horizontal line test or geometric test.

1. If a horizontal line can intersect the graph of the function, more than one time then the function is not mapped as one-to-one.
2. If a horizontal line can intersect the graph of the function only a single time, then the function is mapped as one-to-one.

## One to One Function Inverse

If f is a function defined as y = f(x), then the inverse function of f is x = f -1(y) i.e. f-1 defined from y to x. In inverse function co-domain of f is the domain of f -1 and the domain of f is the co-domain of f -1.

Only one-to-one functions has its inverse since these functions has one to one correspondences i.e. each element from the range correspond to one and only one domain element.

Let a function f : A -> B is defined, then f is said to be invertible if there exists a function g : B -> A in such a way that if we operate f{g(x)} or g{f(x)} we get the starting point or value.

Let us understand with the help of an example,

Example: Show that the function f : X -> Y, such that f(x)= 5x + 7,

For all x , y  ∈ N is invertible.

Solution:

Let y ∈ N -> y = f(x) = 5x + 7 for x ∈ N

x = (y-7)/5

If we define h : Y -> X by h(y) = (y-7)/ 5

Again hof(x) = h[ f(x) ] = h{ 5x + 7 } = 5(y-7) /  5 + 7 = x

And foh(y) = f[ h(y) ] = f( (y-7) / 5) = 5(y-7) /  5 + 7 = y

Hence f is invertible function and h is the inverse of f.

## Properties of One-One Function

• If f and g are both one to one, then f ∘ g follows injectivity.
• If g ∘ f is one to one, then function f is one to one but function g may not be.
• f : X → Y is one-one, if and only if, given any functions g, h : P → X whenever f ∘ g = f ∘ h, then g = h. In other words, one-one functions are exactly the monomorphisms in the category Set of sets.
• If f : X → Y is one-one and P is a subset of X, then f-1 (f(A)) = P. Thus, P can be retrieved from its image f(P).
• If f : X → Y is one-one and P and Q are both subsets of X, then f(P ∩ Q) = f(P) ∩ f(Q).
• If both X and Y are limited with the same number of elements, then f : X → Y is one-one, if and only if, f is surjective or onto function.

## Examples

Example 1: Let A = {1, 2, 3} and B = {a, b, c, d}. Which of the following is a one-to-one function?

1. {(1, c), (2, c)(2, c)}
2. {(1,a),(2,b),(3,c)}
3. {(1, b)(1, c)}

Explanation: Here, option number 2 satisfies the one-to-one condition, as elements of set B(range) is uniquely mapped with elements of set A(domain).

Example 2: Show that f: R→ R defined as f(a) = 3a3 – 4 is one to one function?

Solution:

Let f ( a1 ) = f ( a2 ) for all a1 , a2∈R

so 3a13 – 4 = 3a23 – 4

a13 = a23

a13 – a23 = 0

(a1 – a2) (a1 + a1a2 + a22) = 0

a1 = a2 and (a12 + a1a2 + a22) = 0

(a12 + a1a2 + a22) = 0 is not considered because there is no real values

of a1 and a2.

Therefore,  the given function f is one-one.