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 \( f: A \rightarrow B \) 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: \( f(x) = 3x - 5 \), \( f: \mathbb{R} \rightarrow \mathbb{R} \)
- Check Injectivity (One-to-One):
Assume \( f(a) = f(b) \)
\( 3a - 5 = 3b - 5 \Rightarrow a = b \)
So, the function is injective. - Check Surjectivity (Onto):
Let \( y \in \mathbb{R} \) and \( f(x) = y \)
\( 3x - 5 = y \Rightarrow x = \frac{y+5}{3} \in \mathbb{R} \)
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 \( f(x) = ax + b \) with \( a \neq 0 \), the function is always bijective from \( \mathbb{R} \) to \( \mathbb{R} \).
- For functions like \( f(x) = x^2 \), restrict domain to positive numbers (\( \mathbb{R}^{+} \)), 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 \( f(x) = x^3 \), \( f: \mathbb{R} \rightarrow \mathbb{R} \), bijective?
2. For every \( y \in \mathbb{R} \), \( x = \sqrt[3]{y} \in \mathbb{R} \) (surjective)
3. Conclusion: \( f(x) = x^3 \) is bijective.
Example 2: Is \( f(x) = x^2 \), \( f: \mathbb{R} \rightarrow \mathbb{R} \), bijective?
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 \( f(x) = 2x + 1 \), from \( \mathbb{R} \) to \( \mathbb{R} \), is bijective.
- Find out if \( f(x) = \sin x \), \( f:\mathbb{R} \rightarrow [-1,1] \), is bijective.
- Write the inverse of \( f(x) = 4x-7 \) 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.
Explore related topics: Injective and Surjective Functions, Inverse Functions, Types of Functions, Relations and Functions
FAQs on Bijective Function Explained: Meaning, Proofs & Examples
1. What is a bijective function?
A bijective function, also known as a bijection, is a function where each element in the domain maps uniquely to one element in the codomain, and every element in the codomain is mapped to by exactly one element in the domain. It's both injective (one-to-one) and surjective (onto).
2. How is a bijective function different from injective and surjective functions?
An injective function (one-to-one) ensures each element in the domain maps to a unique element in the codomain, but not all codomain elements need to be mapped to. A surjective function (onto) ensures all codomain elements are mapped to, but domain elements might map to the same codomain element. A bijective function is both injective and surjective, meaning each domain element maps uniquely to one codomain element, and vice-versa.
3. Is x³ a bijective function?
Yes, f(x) = x³ is a bijective function from the set of real numbers (ℝ) to itself. It's both injective (different x values yield different y values) and surjective (every real number y has a corresponding real number cube root x).
4. How do you prove a function is bijective?
To prove a function is bijective, you must prove it's both injective and surjective:
•Injectivity: Show that if f(a) = f(b), then a = b. This means distinct inputs map to distinct outputs.
•Surjectivity: Show that for every element y in the codomain, there exists an element x in the domain such that f(x) = y. This means every element in the codomain has a pre-image in the domain.
5. What are some examples of bijective functions?
Examples include:
• f(x) = x (the identity function)
• f(x) = ax + b (where a ≠ 0)
• f(x) = ex (from ℝ to (0, ∞))
• f(x) = tan-1x (from ℝ to (-π/2, π/2)). Note that the domain and codomain must be carefully defined for bijectivity.
6. Are all invertible functions bijective?
Yes, a function is invertible if and only if it is bijective. The inverse function reverses the mapping, ensuring a one-to-one correspondence between domain and codomain.
7. What is the importance of bijective functions in constructing inverse functions?
Bijective functions are essential for constructing inverse functions. Only bijections possess a well-defined inverse because each element in the codomain has precisely one pre-image in the domain, allowing for a unique reverse mapping.
8. How do bijections apply to set theory and combinatorics?
In set theory, bijections establish a one-to-one correspondence between sets, proving they have the same cardinality (size). In combinatorics, bijections are used to count the number of elements in a set by finding a bijective mapping to a set whose cardinality is already known.
9. What common mistakes do students make when checking for bijectivity?
Students often confuse injectivity and surjectivity. They might only check one condition (injectivity or surjectivity) instead of both. They might also overlook the importance of specifying the precise domain and codomain, as a function can be bijective for one pair of sets but not another.
10. Can a non-numeric function be bijective?
Yes, a function with non-numeric inputs and outputs can be bijective. Consider a function that maps the days of the week to the planets in our solar system (assuming 7 planets). If each day is uniquely paired with a planet and every planet is assigned, the function would be bijective. The key is a unique one-to-one mapping.
11. What are some real-world applications of bijective functions?
Bijective functions are crucial in areas like cryptography (encryption/decryption), computer science (data structures and algorithms), and coding theory (error correction). They enable secure and reversible transformations of data.
