One to One Function
The term one to one relationships actually refers to relationships between any two items in which one can only belong with only one other item. In a mathematical sense, these relationships can be referred to as one to one functions, in which there are equal numbers of items, or one item can only be paired with only one other item. The name of a person and the reserved seat number of that person in a train is a simple daily life example of one to one function.
If you are curious about what makes one to one functions special, then this article will help you learn about their properties and appreciate these functions. Using solved examples, let us explore how to identify these functions based on expressions and graphs.
What Is a One to One Function?
One to One Function Definition
One to one functions are special functions that return a unique range for each element in their domain i.e, the answers never repeat. As an example the function g(x) = x  4 is a one to one function since it produces a different answer for every input. Also, the function g(x) = x2 is not a one to one function since it produces 4 as the answer when the inputs are 2 and 2. A function that is not a one to one is called a many to one function.
Algebraically, we can define one to one function as:
function g: D > F is said to be onetoone if
\(g\left(x_{1}\right)=g\left(x_{2}\right) \Rightarrow x_{1} =x_{2}\)
for all elements \(x_{1}\) and \(x_{2}\) ∈ D. A one to one function is also considered as an injection, and a function is injective only if it is onetoone. The contrapositive of this definition is a function g: D > F is onetoone if \(x_{1} \neq x_{2} \Rightarrow g\left(x_{1}\right) \neq g\left(x_{2}\right)\). Let us visualize this by mapping two pairs of values to compare functions that are and that are not one to one
In the Fig (a), x is the domain and f(x) is the codomain, likewise in Fig (b), x is a domain and g(x) is a codomain
In Fig(1), for each fx value, there is only one unique value of f(x) and thus, f(x) is one to one function.
In Fig (2), different values of x, 2, and 2 are mapped with a common g(x) value 4 and the different x values 4 and 4 are mapped to a common value 16. Thus, g(x) is a function that is not a one to one function.
Properties of One to One Function
A onetoone function i.e an injective function that maps the distinct elements of its domain to the distinct elements of its codomain. Here are some properties that help us to understand the various characteristics of one to one functions:
 If two functions, f(x) and k(x), are one to one, the f ◦ k is a one to one function as well. f ◦ k (\(x_{1}\)) = f ◦ k (\(x_{2}\)) ⇒ f(k(\(x_{1}\))) = f(k(\(x_{2}\))) ⇒ f(\(x_{1}\)) = k(\(x_{2}\)) ⇒ \(x_{1}\) = \(x_{2}\)
 The domain of the function g equals the range of g1 and the range of g equals the domain of g1
 If a function is considered to be one to one, then its graph will either be always increasing or always decreasing.
 g1 (g(x)) = x, for every x in the domain of g, and g(g1(x)) = x for every x in the domain of g1
 If f ◦ k is a one to one function, then k(x) is also guaranteed to be a one to one function
 The graph of a function and the graph of its inverse are symmetric with each other with respect to the line k = j
How to Determine if a Function Is One to One?
Vertical line tests are used to determine if a given relation is a function or not. Further, we can determine if a function is one to one by using two methods:
 Testing one to one function geometrically: If the graph of g(x) passes through a unique value of y every time, then the function is said to be one to one function.
 Testing one to one function algebraically: The function g is said to be one to one if a = b for every g(a) = g(b)
One to One Function Graph
Any function can be represented in the form of a graph. This function is represented by drawing a line on a plane as per the cartesian sytem. The domain is marked horizontally with reference to the xaxis and the range is marked vertically in the direction of the yaxis. If a function g is one to one function then no two points (\(x_{1}\),\(y_{1}\)) and (\(x_{2}\),\(y_{2}\)) have the same yvalue. Therefore no horizontal line cuts the graph of the equation y = g(x) more than once. Here, in this onetoone function shown in the figure, each 'a' is expected to have a unique value of 'y'. Since each 'a' will have a unique value for 'y', one to one functions will never have ordered pairs that share the same ycoordinate.
Inverse of One to One Function
It is essential for one to understand the concept of one to one functions in order to understand the concept of inverse functions and to solve certain types of equations. Firstly, a function g has an inverse function, g1, if and only if g is one to one. In the belowgiven image, the inverse of a onetoone function g is denoted by g−1, where the ordered pairs of g1 are obtained by interchanging the coordinates in each ordered pair of g. Here the domain of g becomes the range of 1, and the range of g becomes the domain of g1.
Properties of the Inverse of One to One Function
The inverse of one to one function undoes what the original function did to a value in its domain in order to get back to the original yvalue. Here are the properties of the inverse of one to one function:
 The function f has an inverse function if and only if f is a one to one function i.e, only one to one functions can have inverses.
 If the functions g and f are inverses of each other then, both these functions can be considered as one to one functions
 If a and b are inverses of each other if and only if (a ◦ b) (x) = x, x in the domain of b and (b ◦ a) (x) = x, x in the domain of a. Here a ◦ b is the composition function that has 'a' composed with 'b'. 'a' and 'b' are called composite functions. In calculus, we use this notation ◦ denote that two functions are combined together to make a new function
 If a and b are inverses of each other then the domain of a is equal to the range of b and the range of a is equal to the domain of b.
 If a and b are inverses of each other then their graphs are will make reflections of each other on the line y = x.
 If the point (c,d) is on the graph of f then point (d,c) is on the graph of f1.
Steps to Find the Inverse of One to One Function
The step by step procedure to derive the inverse function g^{1}(x) for a one to one function g(x) is as follows: Set g(x) equal to y
 Switch the x with y since every (x,y) has a (y,x) partner
 Solve for y
 In the equation just found, rename y as g^{1} (x).
Example: Find the inverse function g^{1}(x) of the function g(x) = 2 x + 5.
Now, let us follow the 4 steps:
Set g(x) equal to y  y = 2x + 5 
Switch x with y  x = 2y + 5 
Solve for y  y = (x5)/2 
Rename y as g^{}1(x). This is the inverse.  g^{1}(x) = (y5)/2 
Related Articles
Check out the following pages related to one to one function
 Introduction to the Concept of Functions
 What are Functions?
 Inverse of a Function
 Graphing Functions
 Linear Functions
Important Notes on One to One Function
Here is a list of a few points that should be remembered while studying one to one function:
 In a mathematical sense, one to one functions are functions in which there are equal numbers of items in the domain and in the range, or one can only be paired with another item.
 It is essential for one to understand the concept of one to one functions in order to understand the concept of inverse functions and to solve certain types of equations.
 One can easily determine if a function is one to one geometrically and algebraically too.
Examples on One to One Function

Example 1: Let D = {3, 4, 8, 10} and C = {w, x, y, z}. Which of the following sets of ordered pairs represent a one to one function?
{(3, w), (3, x), (3, y), (3,z)}
{(4,w), (3,x), (10,z), (8, y)}
{(4,w), (3,x), (8,x), (10, y)}Solution:
For a function to be a one to one function, each element from D must pair up with a unique element from C.
 In the first option, the same value of x is mapped with each value of y, so it cannot be considered as a function and, hence it is not a onetoone function.
 In the third option, different values of x are mapped with each ordered pair, but 3 and 8 share the same range of x. Hence, it is not a one to one function.
 The second option uses a unique element from D for every unique element from C, thus representing a onetoone function.
This means that {(4,w), (3,x), (10,z), (8, y)} represent a one to one function.

Example 2: Determine if g(x) = 3x^{3} – 1 is a one to one function using the algebraic approach.
Solution:
In order for function to be a one to one function, g( \(x_{1}\) ) = g( \(x_{2}\) ) if and only if \(x_{1}\) = \(x_{2}\) . Let us start solving now:
g( \(x_{1}\) ) = 3 \(x_{1}\) 3 – 1
g( \(x_{2}\) ) = 3 \(x_{2}\) 3 – 1
Equate both expressions and see if it reduces to \(x_{1}\) = \(x_{2}\) .
3 \(x_{1}\) 3 – 1 = 3 \(x_{2}\) 3 – 1
3 \(x_{1}\) 3 = 3 \(x_{2}\) 3
( \(x_{1}\) )3 = ( \(x_{2}\) )3
Removing the cube roots from both the sides of the equation will lead us to \(x_{1}\) = \(x_{2}\). Hence, g(x) = 3x3 – 1 is a one to one function.
FAQs on One to One Function
What Is a One to One Function?
One on one functions are special functions that return a unique range for each element in their domain i.e, the answers never repeat. A normal function can actually have two different input values that can produce the same answer, whereas a one to one function does not.
How Do You Check if a Function Is One to One?
One can check if a function is one to one by using either of these two methods:
 Testing one to one function geometrically: If the graph of g(y) passes through a unique value of x every time, then the function can be considered as a one to one function.
 Testing one to one function algebraically: The function g is said to be one to one if x = y for every g(x) = g(y)
What Types of Functions Are One to One Functions?
A one to one function is either strictly decreasing or strictly increasing. In a one to one function, the same values are not assigned to two different domain elements
What Does It Mean if a Function Is Not One to One Function?
In a function, if a horizontal line passes through the graph of the function more than once, then the function is not considered as onetoone function. Also,if the equation of x on solving has more than one answer, then it is not a one to one function. A function that is not a one to one is considered as many to one.
What Are the Steps in Solving the Inverse of a One to One Function?
These are the steps in solving the inverse of a one to one function:
 Set g(x) equal to y
 Solve the equation y = g (x) for x. If there is only one solution then the inverse can exist; otherwise, it can't.
 In the equation just found, rename x as g1 (y).
What Is an Example of a One to One Function?
The function f(x) = x + 5 is a one to one function as it produces different output for a different input x. And for a function to be one to one it must return a unique range for each element in its domain. Here, f(x) returns 6 if x is 1, 7 if x is 2 and so on. A person and his shadow is a reallife example of one to one function.
What Is Not a One to One Function?
The function f(x) = x2 is not a one to one function as it produces 9 as the answer when the inputs are 3 and 3. And for a function to be one to one it must return a unique range for each element in its domain. Here, f(x) returns 9 as an answer, for two different input values of 3 and 3.
Are Parabolas One to One Functions?
No, parabolas are not one to one functions. The function g(y) = y2 is not onetoone function because g(2) = g(2). The function g(y) = y2 graph is a parabolic function, and many horizontal lines pass through the parabola twice.
How Can We Apply the Concept of One to One Function in Daily Life?
We can see these one to one relationships everywhere. One of the very common examples of a one to one relationship that we see in our everyday lives is where one person has one passport for themselves, and that passport is only to be used by this one person.