One to One Function

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. With the help of examples, we are going to learn about this function in detail so that its concept could be easily understood. 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

Also, we have other types of functions in Maths which you can learn here easily, such as Identity function, Constant function, Polynomial function, etc. Let us now learn, a brief explanation with definition, its representation and example.+

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 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

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,

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

For all x , y  ∈ N is an 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.

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)}

The Answer is 2.

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 \(\rightarrow\) R defined as f(a) = 3a3 – 4 is one to one function?

Solution:

Let f ( a1 ) = f ( a2 ) for all a1 , a2 \(\in\) 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.

Students are advised to solve more of such example problems, to understand the concept of one-to-one mapping clearly. To learn more about various maths concepts, register with BYJU’S. Also, download its app to get personalised learning videos.

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