One-to-One and Onto Functions – Meaning, Differences & Examples
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Functions are an important concept in mathematics that are used to describe relationships between sets. They allow us to model and analyze various phenomena in the real world, from the movement of planets to the behavior of financial markets. One of the most important properties of a function is whether it is one-to-one and onto, as these properties have significant implications for the behavior of the function.
In this mathematics article, we will explore what it means for a function to be one-to-one and onto, one-to-one and onto function examples, and why understanding these properties is crucial in many mathematical fields.
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One-to-One Functions
A one-to-one function is a type of function that maps each element in the range to only one element in its domain, which means that the outputs are unique and never repeated.
What is One-to-One and Onto Function?
A function is said to be both one-to-one and onto if it satisfies both properties. In other words, each element in the domain is paired with a unique element in the range, and every element in the range is mapped to by at least one element in the domain.
Graphically, a one-to-one and onto function is a function where every point in the range is covered by exactly one point in the domain, and no horizontal line intersects the graph more than once.
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How to Check One-to-One and Onto Function?
One-to-One and Onto Function Graph
To provide a visual understanding of one-to-one and onto functions, here are the graphical representations of each:
One-to-One Function Graph:
The horizontal line test is used to determine whether a function is one-one when its graph is given. to test whether the function is one-one from its graph,
- To determine if a function is one-to-one, imagine a horizontal line (like a stick) passing through the graph.
- If the line never intersects more than one point on the graph at any given moment, then the function is one-to-one.
- However, if the line does intersect multiple points of the graph at any instance, then the function is not one-to-one.
For example, Consider the graph below.
Onto Function Graph:
To determine if a function is onto using its graph, a straightforward approach is to compare the range with the codomain. If the range is equal to the codomain, then the function is onto. For any function graph, it can be considered onto only if every horizontal line intersects the graph at one or more points. If there exists an element in the range that fails to intersect the function's graph when subjected to the horizontal line test, then the function is not onto. The following image exemplifies a graph of an onto function:
Difference Between One-to-One and Onto Function
Here's a tabular format highlighting the key differences between one-to-one and onto functions:
One-to-One and Onto Function Solved Examples
1.Determine if the function
Solution:
To check if
Since
To check if
Therefore, f(x) is onto.
2.Determine if the function
Solution:
To check if
Since
To check if
Therefore,
3.Determine if the function
Solution:
To check if
Since
To check if
Therefore,
Properties of a One-One Function
-
Composition Rule:
Composition Rule:
If both functions f and g are one-one (injective), then their combination f ∘ g is also one-one.
-
Reverse Rule:
If g ∘ f is one-one, then f must be one-one. But g may or may not be one-one. -
Unique Mapping:
A function f: X → Y is one-one if, for any two functions g and h (from a set P to X),
if f ∘ g = f ∘ h, then it means g = h.
This shows that one-one functions are special because they preserve uniqueness. -
Image and Pre-image:
If f: X → Y is one-one and P is a part (subset) of X,
then we can get back P from its image: f⁻¹(f(P)) = P -
Intersection Rule:
If f is one-one and P and Q are two subsets of X, then: f(P ∩ Q) = f(P) ∩ f(Q) -
Finite Set Rule:
If X and Y have the same number of elements (finite sets),
then a function f: X → Y is one-one if and only if it is also onto (surjective).
We hope that the above article is helpful for your understanding and exam preparations. Stay tuned to the Testbook App for more updates on related topics from Mathematics and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams. For better practice, solve the below provided previous year papers and mock tests for each of the given entrance exam:
FAQs for One-to-One and Onto Function
What is the difference between a one-to-one function and an onto function?
A one-to-one function maps each element of the domain to a unique element of the range, ensuring that no two different inputs produce the same output. On the other hand, an onto function maps every element in the range to at least one element in the domain, guaranteeing that no elements are left unmapped in the range.
Can a function be both one-to-one and onto?
Yes, a function can be both one-to-one and onto. In fact, such functions are referred to as bijections.
How can I determine if a function is one-to-one or onto?
To determine if a function is one-to-one, you can use the horizontal line test. If no horizontal line intersects the graph of the function more than once, then the function is one-to-one. On the other hand, to verify if a function is onto, you need to check if every element in the range has a corresponding element in the domain such that the function maps to it.
What are the applications of one-to-one and onto functions?
One-to-one and onto functions have various applications in mathematics and other fields. One-to-one functions are useful in solving equations, cryptography, computer science, and modeling consumer preferences and demand. Onto functions find applications in physics, statistics, data analysis, and coding theory, among others.
Can a function be neither one-to-one nor onto?
Yes, it is possible for a function to be neither one-to-one nor onto. Such functions may have elements in the domain that map to the same element in the range (not one-to-one), and/or there may be elements in the range that are not mapped to by any element in the domain (not onto).
Define one-to-one and onto function.
A function is said to be both one-to-one and onto if it satisfies both properties. In other words, each element in the domain is paired with a unique element in the range, and every element in the range is mapped to by at least one element in the domain
How do you know if a function is one-to-one?
If no two inputs give the same output, the function is one-to-one. You can also check it using the horizontal line test on a graph.
How do you know if a function is onto?
If every value in the output set is used at least once by the function, then it's onto.
Why are one-to-one and onto functions important?
These functions help us understand mappings between sets and are used in algebra, calculus, and computer science.
How are these functions used in real life?
They’re used in coding, database mapping, cryptography, and solving equations where precise input-output matching is needed.