One to One Functions in Mathematics

A One-to-One function, also known as an Injective function, is a type of function defined over a domain and codomain that describes a specific type of relationship between them. In a One-to-One function, each element in the domain maps to a unique element in the codomain.

This article explores the concept of One to One Function or One-One Function in detail, including its definition and examples, which help you understand the concept with ease.

One-to-One Function Mathematical Definition

A function 'f' from a set 'A' to a set 'B' is one-to-one if no two elements in 'A' are mapped to the same element in 'B.'

Let's consider these two diagrams. For diagram A, we realize that 10 maps to 1, 20 maps to 2, and 30 maps to 3.

However for diagram B it is clear that 10 and 30 maps to 3 and then 20 maps to 1.

Since we have elements in the domain corresponding to distinct values in the each domain for diagram A it makes the function one-to-one, thus our diagram B is not one to one.

This can be expressed mathematically as

f(a) = f(b) ⇒ a = b

One-to-One Functions Example

Some examples of one-to-one functions are given below:

• Identity Function: The identity function is a simple example of a one-to-one function. It takes an input and returns the same value as the output. For any real number x, the identity function is defined as:

f(x) = x

Every distinct input x corresponds to a distinct output f(x), making it a one-to-one function.

• Linear Function: A linear function is one where the highest power of the variable is 1. For example:

f(x) = 2x + 3

This is a one-to-one function because no matter what value of x you choose, you will get a unique value for f(x).

• Square Function: The square function, defined on positive real numbers, f(x)= x^2, is also a one-to-one function. For any positive real number x, the square value function returns a non-negative value, and different values of x will result in different values.

Let's prove one such example for a one-to-one function.

Example: Prove that the function f(x) = 1/(x+2), x≠2 is one-to-one.

Solution:

According one-to-one function we know that
f(a) = f(b)

replace a with x and x with b
f(a) = 1/(a+2) , f(b) = 1/(b+2)
⇒ 1/(a+2) = 1/(b+2)

cross multiply the above equation
1(b+2)=1(a+2)
b+2=a+2
⇒ b=a+2-2

∴ a=b

Now, since a = b the function is said to be one-to-one function.

Properties One-to-One Functions

Let's consider that f and g are two one-to-one functions, the properties are as follows:

• 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 of sets.
• If f: X → Y is one-to-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-to-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-to-one, if and only if f is a surjective or onto function.

One-to-One Function Graph

Let's see one of a graph representation of the one-to-one function

The above graph of the function f(x)= √x shows the graphical representation of a one-to-one function.

Horizontal Line Test

A function is one-to-one if each horizontal line does not intersect the graph at more that one point.

In the above example, it only intersect the horizontal line only at one point. So f(x) is one-to-one function which means that it has an inverse function.

Inverse of a one-to-one function

Let f be a one-to-one function with a domain A and Range B. Then the inverse of f is a function with domain B and Range A defined by f^-1 (y) =x if and only if f(x) = y for any y in B.

Always remember a function has an inverse if and only if it is one-to-one. A function is one-to-one if the highest exponent is an odd number. But if the highest number is an even number or an absolute value this is not one-to-one function.

Example: f(x) = 3x + 2. Find the inverse of the function.
Solution:

write the function in y=f(x) form
⇒ y = 3x + 2

lets interchange y and x variables
⇒ x = 3y + 2

solve y in terms of x
⇒ x - 2=3y

divide the equation with 3
⇒ (x-2)/3=3y/3
⇒ y=(x-2)/3

∴ f^-1(x)=(x-2)/3

One-to-One Function and Onto Function

The key differences between One-to-One and Onto Functions are listed in the following table:

The following illustration provides the clear difference between one-to-one and onto functions:

Read More,

• Functions
• Types of Functions
• Relation and Function

Solved Problems on One-to-One Function

Let's solve some problems to illustrate one-to-one functions:

Question 1: Determine if the following function is one-to-one: f(x) = 3x - 1.
Solution:

To check if it's one-to-one, we need to show that no two distinct x-values map to the same y-value.

Suppose f(a) = f(b), where a ≠ b.
3a - 1 = 3b - 1
3a = 3b
a = b

Since the only way for f(a) = f(b) is when a = b, this function is indeed one-to-one.

Question 2: Determine if the following function is one-to-one: g(x) = x²
Solution:

We'll use the horizontal line test by graphing the function. If any horizontal line intersects the graph more than once, it's not one-to-one.

The graph of g(x) = x^2 is a parabola opening upwards. Any horizontal line only intersects the graph once, so this function is not one-to-one.

Practice Problems on One-to-One Functions

Question 1: Determine whether the following function is one-to-one:

• f(x) = 2x + 3
• g(x) = 3x² - 1
• h(x) = ³√x

Question 2: Find a function that is one-to-one from the set of real numbers to the set of real numbers.

Question 3: Given the function g(x) = x² + 1, determine if it is one-to-one on its entire domain.

Question 4: Consider the function h(x) = e^x. Is it a one-to-one function?

Question 5: Find the inverse function of f(x) = 4x - 7 and determine its domain.

Question 6: Determine if the function p(x) = √x is one-to-one.

Question 7: Given q(x) = x/2, find the domain and range of the function.

Question 8: Check whether the function r(x) = sin (x) is one-to-one over the interval [0, π].

Question 9: Consider the function s(x) = |x|. Is it a one-to-one function?

Question 10: Determine if the function t(x) = 1/x is one-to-one and find its domain.