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Bijective Function Explained: Meaning, Proofs & Examples

Bijective Function Explained: Meaning, Proofs & Examples

Q: 1. What is the definition of a bijective function as per the Class 12 syllabus?

A function is defined as bijective (or a bijection) if it is both injective (one-to-one) and surjective (onto). This means that every element in the domain is paired with exactly one unique element in the codomain, and every element in the codomain has exactly one pre-image in the domain, creating a perfect one-to-one correspondence between the two sets.

Q: 6. How can the Horizontal Line Test help determine if a function is bijective?

The Horizontal Line Test is a graphical method used to check for injectivity, which is one of the two conditions for bijectivity. If any horizontal line drawn on the graph of a function intersects the curve at more than one point, the function is not injective and therefore cannot be bijective. However, this test alone is not sufficient to prove bijectivity; you must also confirm surjectivity by checking if the function's range is equal to its entire specified codomain.

Q: 8. What is the most common mistake made when checking for bijectivity in exams?

The most common mistake is only checking one of the two required conditions. Students often prove a function is injective (one-to-one) and incorrectly assume it must also be surjective, or vice versa. Another frequent error is ignoring the specified domain and codomain. A function's bijectivity depends entirely on these sets; for example, f(x) = x² is bijective if its domain and codomain are restricted to non-negative real numbers, but not if they are all real numbers.

Reviewed by:

Rama Sharma

How to Prove a Function is Bijective: Steps with Examples

The concept of bijective function plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding bijective functions helps students master mapping, inverse functions, and function proofs, which are important for Class 12, JEE, and Olympiad exams.

What Is Bijective Function?

A bijective function is defined as a function that is both injective (one-to-one) and surjective (onto). This means every element in the domain maps to a unique element in the codomain, and every element in the codomain is used. You’ll find this concept applied in areas such as inverse functions, relations and mappings, and set theory.

Key Formula for Bijective Function

Here’s the standard formula: A function $f : A \to B$ is bijective if for every $b \in B$ , there is exactly one $a \in A$ such that $f (a) = b$ . In other words, f is injective and surjective.

Cross-Disciplinary Usage

The bijective function is not only useful in Maths but also plays an important role in Physics, Computer Science, cryptography, and logic. Students preparing for JEE or NEET often see bijection in combinatorics questions, coding, and even relatability with function inverses in chemistry kinetics.

Step-by-Step Illustration: How to Prove a Function Is Bijective

Suppose you are given: $f (x) = 3 x - 5$ , $f : R \to R$
Check Injectivity (One-to-One):
Assume $f (a) = f (b)$
$3 a - 5 = 3 b - 5 \Rightarrow a = b$
So, the function is injective.
Check Surjectivity (Onto):
Let $y \in R$ and $f (x) = y$
$3 x - 5 = y \Rightarrow x = \frac{y + 5}{3} \in R$
Every y can be hit, so the function is surjective.
Conclusion: Since it is both one-to-one and onto, $f (x)$ is bijective.

Speed Trick or Quick Visual Test

A fast way to check if a function is bijective in exams:

For linear functions $f (x) = a x + b$ with $a \neq 0$ , the function is always bijective from $R$ to $R$ .
For functions like $f (x) = x^{2}$ , restrict domain to positive numbers ( $R^{+}$ ), and check if every output is used and comes from a unique input.

Vedantu teachers use these speed tricks in live classes for fast last-minute revision and confidence in MCQs.

Bijective vs Injective vs Surjective

Function Type	Meaning	Visual
Injective	Every element in codomain is mapped by at most one element from domain (may not use all codomain).	No two arrows land at same codomain point.
Surjective	Every codomain element is used (may have multiple domain elements map to same output).	All codomain points get at least one arrow.
Bijective	Both injective and surjective—perfect matching, each codomain element comes from one unique domain element.	One-to-one arrows, nothing left unused.

Example Problems on Bijective Function

Example 1: Is $f (x) = x^{3}$ , $f : R \to R$ , bijective?

1. Assume

f (a) = f (b)

a^{3} = b^{3} \Rightarrow a = b

(injective)

2. For every

y \in R

x = \sqrt[3]{y} \in R

(surjective)

3. Conclusion:

f (x) = x^{3}

is bijective.

Example 2: Is $f (x) = x^{2}$ , $f : R \to R$ , bijective?

f (2) = 4 = f (- 2)

→ not injective.
2. Negative outputs (

y = - 1

) not possible; not surjective.
3. Conclusion: Not bijective.

Properties of Bijective Function

Every bijective function has an inverse, which is also a function.
If sets A and B are finite and |A| = |B| = n, then there are n! bijections.
Bijective mappings are used in counting, coding, and constructing reversible algorithms.
In Maths, bijections make function tables “fully matched” — all rows and columns are paired up with no gaps or overlaps.

Try These Yourself

Show $f (x) = 2 x + 1$ , from $R$ to $R$ , is bijective.
Find out if $f (x) = \sin x$ , $f : R \to [- 1, 1]$ , is bijective.
Write the inverse of $f (x) = 4 x - 7$ if it’s bijective.
Draw a mapping diagram for a bijective function between sets A = {1,2,3} and B = {a,b,c}.

Frequent Errors and Misunderstandings

Confusing surjective with bijective—remember, bijection requires both “one-to-one” and “onto”.
Missing domain or codomain restrictions (e.g., with quadratics, cube roots).
Forgetting that inverse exists only for bijections.

Relation to Other Concepts

The idea of bijective function connects closely with topics such as inverse functions, types of functions, and domain and range. Mastering this concept also helps with proof writing and advanced theorems in calculus and algebra.

Classroom Tip

A simple way to remember bijective function: Draw arrows from every domain element to codomain, make sure no arrow shares an endpoint. If all are used once—and only once—you have a bijection! Vedantu’s teachers use such diagrams in classes for fast recognition.

We explored bijective function—from definition, formula, examples, mistakes, and connections to other subjects. Continue practicing with Vedantu to become confident in spotting and using bijective functions.

FAQs on Bijective Function Explained: Meaning, Proofs & Examples

1. What is the definition of a bijective function as per the Class 12 syllabus?

A function is defined as bijective (or a bijection) if it is both injective (one-to-one) and surjective (onto). This means that every element in the domain is paired with exactly one unique element in the codomain, and every element in the codomain has exactly one pre-image in the domain, creating a perfect one-to-one correspondence between the two sets.

2. How can you differentiate between injective, surjective, and bijective functions?

The key difference lies in the mapping conditions they must satisfy:

Injective (One-to-One): Different elements in the domain must have different images in the codomain. It's possible for some elements in the codomain to not be an image of any element from the domain.
Surjective (Onto): Every element in the codomain must be an image of at least one element in the domain. It's possible for multiple domain elements to map to the same codomain element.
Bijective (One-to-One and Onto): This combines both conditions. Every element in the codomain is the image of exactly one element from the domain.

3. What are the essential steps to prove a function is bijective with an example?

To prove a function f: A → B is bijective, you must prove two separate conditions:

1. Prove Injectivity: Assume f(x₁) = f(x₂) for any x₁, x₂ in the domain A. Through algebraic manipulation, show that this implies x₁ = x₂.
2. Prove Surjectivity: Let 'y' be an arbitrary element in the codomain B. Set f(x) = y and solve for x in terms of y. Show that for any 'y' in B, the calculated 'x' exists and is in the domain A.

For example, to prove f(x) = 2x + 3 is bijective from ℝ to ℝ, you would first prove injectivity (2x₁ + 3 = 2x₂ + 3 ⇒ x₁ = x₂) and then surjectivity (let y = 2x + 3, so x = (y-3)/2, which is a real number for any real y).

4. What are some key examples of bijective functions from the NCERT syllabus?

Common examples of bijective functions that Class 12 students should know include:

Any linear function f(x) = ax + b, where a ≠ 0, with its domain and codomain as the set of all real numbers (ℝ).
The cubic function f(x) = x³, where the domain and codomain are ℝ.
The identity function f(x) = x.
Trigonometric functions with restricted domains, such as f(x) = sin(x) with its domain restricted to [-π/2, π/2] and codomain to [-1, 1].

5. Why is it necessary for a function to be bijective to be invertible?

A function must be bijective to ensure its inverse is also a valid function. Here's why:

The injective (one-to-one) property guarantees that each output has a unique input. When you reverse the mapping, this ensures the inverse does not assign multiple outputs to a single input, which would violate the definition of a function.
The surjective (onto) property guarantees that every element in the codomain has a corresponding element in the domain. This ensures that the domain of the inverse function is well-defined and covers the entire original codomain.

Without both properties, the inverse relation would not be a proper function.

6. How can the Horizontal Line Test help determine if a function is bijective?

The Horizontal Line Test is a graphical method used to check for injectivity, which is one of the two conditions for bijectivity. If any horizontal line drawn on the graph of a function intersects the curve at more than one point, the function is not injective and therefore cannot be bijective. However, this test alone is not sufficient to prove bijectivity; you must also confirm surjectivity by checking if the function's range is equal to its entire specified codomain.

7. Why is the function f(x) = x² not bijective on the set of real numbers (ℝ)?

The function f(x) = x² with its domain and codomain as ℝ is not bijective for two reasons:

It is not injective: Different inputs can produce the same output. For example, f(2) = 4 and f(-2) = 4. Since two different domain elements (2 and -2) map to the same codomain element (4), it fails the one-to-one condition.
It is not surjective: The range of f(x) = x² is [0, ∞), which includes only non-negative real numbers. However, the codomain is specified as all real numbers (ℝ). Since negative numbers in the codomain (e.g., -1) have no corresponding pre-image in the domain, it fails the onto condition.

8. What is the most common mistake made when checking for bijectivity in exams?

The most common mistake is only checking one of the two required conditions. Students often prove a function is injective (one-to-one) and incorrectly assume it must also be surjective, or vice versa. Another frequent error is ignoring the specified domain and codomain. A function's bijectivity depends entirely on these sets; for example, f(x) = x² is bijective if its domain and codomain are restricted to non-negative real numbers, but not if they are all real numbers.

9. What is the importance of bijective functions outside of pure mathematics?

Bijective functions are fundamental to many real-world applications. In cryptography, they are used to create encryption and decryption algorithms, where a plaintext message is transformed into ciphertext (encryption) and must be uniquely reversible back to the original plaintext (decryption). In computer science, they are essential for creating hash functions and ensuring data integrity and reversible data compression.

10. If f and g are two bijective functions, is their composition (g∘f) also guaranteed to be bijective?

Yes. If two functions, f: A → B and g: B → C, are both bijective, their composite function g∘f: A → C is also bijective. The composition preserves both the injective and surjective properties. Since f is injective, no two elements in A map to the same element in B. Since g is also injective, these distinct elements in B will then map to distinct elements in C. Similarly, the onto property is maintained, ensuring the final composition is a perfect one-to-one correspondence from set A to set C.