How is a Surjective (Onto) Function Different from an Injective Function?
The concept of surjective function is essential in mathematics and helps in solving real-world and exam-level problems efficiently. Understanding surjective functions forms a strong foundation for higher-level topics in functions and relations, making it vital for students preparing for board exams and competitive tests. Vedantu ensures students develop a deep and practical understanding of such concepts to build confidence in maths.
Understanding Surjective Function
A surjective function, also known as an onto function, is a mapping where every element in the codomain has at least one pre-image in the domain. In simple words, if each value in the codomain (output set) is “hit” or “covered” by the function from the domain, the function is surjective. This concept is widely used in function theory, set mappings, and algebraic relations. It is important in topics such as functions and its types, relations and functions, and mappings between sets.
What Does Surjective Function Mean?
In simple terms, a surjective function means:
Every output in the codomain comes from at least one input in the domain.
No element in the codomain is left out.
A surjective function is also called an "onto function".
The range and codomain are exactly the same in a surjective function.
Surjective, Injective, and Bijective: Quick Comparison
Function types can be confusing. The table below will help you see the difference clearly for your exams:
| Type | Definition | Nickname | Covers Every Output? |
|---|---|---|---|
| Injective | Each output has one unique input | One-to-one | No (may not cover all) |
| Surjective | Every output is mapped by some input | Onto | Yes |
| Bijective | Both injective and surjective | One-to-one onto | Yes, and each is unique |
Formula Used in Surjective Function
The standard definition for surjectivity is:
If \( f: A \rightarrow B \), then f is surjective if for every \( b \in B \), there exists at least one \( a \in A \) such that \( f(a) = b \).
Worked Example – How to Check if a Function is Surjective
Let’s solve a step-by-step example:
1. Consider \( f: \mathbb{R} \rightarrow \mathbb{R} \) defined by \( f(x) = 2x + 3 \). Is it surjective?2. Set \( f(x) = y \), so \( y = 2x + 3 \).
3. Solve for x:
Subtract 3: \( y - 3 = 2x \)
Divide by 2: \( x = \frac{y - 3}{2} \)
4. For every real number y, x is a real number.
5. Therefore, for every y in the codomain, there is a real x in the domain.
6. So, f(x) = 2x + 3 is a surjective function from \(\mathbb{R}\) to \(\mathbb{R}\).
Now try this one: Is \( f(x) = x^2 \) surjective from \( \mathbb{R} \) to \( \mathbb{R} \)?
For any negative y, \( x^2 = y \) has no real solution (since square of any real number is non-negative). So, this function is not surjective on these sets.
More Practice Problems
- Show that the function \( f: \mathbb{Z} \rightarrow \mathbb{Z} \), \( f(x) = x+1 \), is a surjective function.
- Is \( f(x) = e^x \) surjective from \( \mathbb{R} \) to \( \mathbb{R} \)?
- Given sets A = {1, 2, 3}, B = {a, b}, define a surjective function from A to B.
- Check if \( f(x) = |x| \) from \(\mathbb{R}\) to \([0, \infty)\) is a surjective function.
Common Mistakes to Avoid
- Forgetting that the codomain matters, not just the set of actual outputs (range).
- Assuming a function is surjective just because every input maps somewhere.
- Mixing up surjective (onto) with injective (one-to-one) functions.
- Not checking solutions for all elements of the codomain.
When is a Function Surjective? (Shortcuts & Visuals)
A function is surjective if:
- The range equals the codomain.
- No element in the codomain is left unmapped.
- In mapping diagrams, every output has at least one incoming arrow from the domain.
For real functions, check if for every y in the codomain you can solve for x in the domain.
Visual representations, like mapping diagrams or function graphs that “hit” the whole output set, can help you understand surjective functions better. For practice, check out function graph examples on Vedantu.
Real-World Applications of Surjective Function
Surjective functions play a role in assigning resources, scheduling events, cryptography, and modeling any process where all possible outcomes must be accounted for. Vedantu’s step-by-step explanations make it easy for students to connect the mathematical idea to real-life scenarios.
We explored the idea of surjective function, how to identify and prove surjectivity, solve related problems, and see how the concept works in real life. Practice more with Vedantu and keep exploring related topics for complete exam preparation.
Related Vedantu Pages
- Onto Function
- Bijective Function
- Functions and Its Types
- Relations and Functions
- Domain and Range Relations
- Differences Between Codomain and Range
- JEE Main Sets, Relations & Functions Revision Notes
- Sets, Relations & Functions Practice Paper
- Important Questions on Functions
- Relations & Functions Worksheet
FAQs on What Is a Surjective Function? Definitions and Key Differences
1. What is a surjective function?
Surjective function (also known as an onto function) is a type of function in which every element in the codomain (target set) has at least one pre-image in the domain (origin set). In other words, a function f: A → B is surjective if for every b in B, there exists at least one a in A such that f(a) = b.
2. What is the difference between injective and surjective functions?
Injective function means each element in the codomain is mapped by at most one element from the domain (no two domain elements map to the same codomain element). Surjective function means every element in the codomain is mapped by at least one element from the domain. An injective function is also called one-to-one, while a surjective function is also called onto.
3. Why is an onto function called surjective?
An onto function is called surjective because it ensures the entire codomain is 'covered' by the function. The term surjective comes from the Latin ‘sur’ (meaning over) and ‘jacere’ (to throw), meaning the function throws its domain over the entire codomain so every element has a pre-image.
4. How to know if a function is surjective?
To check if a function is surjective:
1. Examine whether every element in the codomain is mapped by at least one element from the domain.
2. Mathematically, for a function f: A → B and for every b in B, there must exist an a in A such that f(a) = b.
3. For example, check if the equation f(a) = b can be solved for every b in the codomain.
5. Give an example of a surjective function.
An example of a surjective function:
Let f: ℝ → ℝ be defined as f(x) = 2x + 1.
For every y in ℝ, you can find x = (y - 1)/2 in ℝ such that f(x) = y, so f is surjective.
6. What is the definition of an injective function?
An injective function (or one-to-one function) is a function f: A → B in which different elements in the domain map to different elements in the codomain, that is, if f(a1) = f(a2), then a1 = a2.
7. What is a bijective function?
A bijective function is both injective (one-to-one) and surjective (onto). This means every element of the codomain is mapped to by exactly one element of the domain, and there are no unused elements in either set.
8. Surjective function is also called?
Surjective function is also called an onto function.
9. What is the formula to check surjectivity?
The condition for surjectivity is:
For all y in codomain B, there exists x in domain A such that f(x) = y.
This is usually written as: ∀ y ∈ B, ∃ x ∈ A, f(x) = y.
10. Provide a non-surjective function example.
Let f: ℕ → ℕ be defined as f(x) = x + 1. The number 0 in the codomain is not mapped from any element in the domain, so this function is not surjective.
11. Surjective function vs injective function: what's the key difference?
The key difference is:
Injective: No codomain element is the image of more than one domain element (one-to-one, but not necessarily onto).
Surjective: Every codomain element is at least the image of one domain element (onto, but not necessarily one-to-one).
Bijective means both properties hold.
12. Practice problem: Is the function f(x) = x2 from ℝ → ℝ surjective?
No, the function f(x) = x2 from ℝ → ℝ is not surjective, because negative real numbers do not have a real square root, so not every element in the codomain ℝ is covered.
