Example of One-to-One Functions

• 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).

• Absolute Value Function: The absolute value function f(x)=∣x∣ is also a one-to-one function. For any real number x, the absolute value function returns a non-negative value, and different values of x will result in different absolute values.

Let’s prove one such examples for 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.

One to One Function or One-One Function is one of the types of functions defined over domain and codomain and describes the specific type of relationship between domain and the codomain. One to One Function is also called the Injective Function. One to One Function is a mathematical function where 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. We will also discuss some sample problems and provide some practice problems for you to solve. So, let’s learn about this important concept in mathematics known as One to One Function.