What Is a Bijective Function Definition Formula Properties and Solved 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 Complete Guide with Definition and Examples
1. What is a bijective function?
A bijective function is a function that is both one-to-one (injective) and onto (surjective), meaning every element in the codomain has exactly one pre-image in the domain.
- Injective: Different inputs give different outputs.
- Surjective: Every element in the codomain is mapped by some input.
- This ensures a perfect pairing between domain and codomain elements.
2. What does it mean for a function to be one-to-one and onto?
A function is one-to-one and onto if it is both injective (no two inputs share the same output) and surjective (every element of the codomain is covered).
- Injective condition: If f(a) = f(b), then a = b.
- Surjective condition: For every y in codomain, there exists x such that f(x) = y.
3. How do you check if a function is bijective?
To check if a function is bijective, you must prove it is both injective and surjective.
- Step 1: Test injectivity by assuming f(a) = f(b) and show a = b.
- Step 2: Test surjectivity by solving f(x) = y for x and showing a solution exists for all y in the codomain.
- If both conditions hold, the function is bijective.
4. What is an example of a bijective function?
An example of a bijective function is f(x) = 2x + 3 from ℝ to ℝ.
- It is injective because if 2a + 3 = 2b + 3, then a = b.
- It is surjective because for any y, solving y = 2x + 3 gives x = (y − 3)/2.
- Therefore, f(x) is bijective.
5. Does a bijective function always have an inverse?
Yes, a function has an inverse if and only if it is bijective.
- Because each output corresponds to exactly one input.
- The inverse function reverses the mapping.
- For example, if f(x) = 2x + 3, then f⁻¹(x) = (x − 3)/2.
6. What is the difference between injective, surjective, and bijective functions?
The difference lies in how elements of the domain and codomain are related.
- Injective (one-to-one): No repeated outputs.
- Surjective (onto): Every codomain element is mapped.
- Bijective: Both injective and surjective.
7. Can a quadratic function be bijective?
A quadratic function is not bijective over ℝ to ℝ, but it can be bijective if its domain is restricted.
- For example, f(x) = x² is not injective because f(2) = f(−2).
- If the domain is restricted to x ≥ 0, then it becomes injective.
- With domain x ≥ 0 and codomain y ≥ 0, it is bijective.
8. Why are bijective functions important?
Bijective functions are important because they guarantee a one-to-one correspondence and always have an inverse.
- They allow reversible mappings.
- They are essential in algebra, calculus, and discrete mathematics.
- They establish equal cardinality between sets.
9. What is the formula for the inverse of a bijective function?
The inverse of a bijective function is found by solving y = f(x) for x and then writing x in terms of y.
- Step 1: Replace f(x) with y.
- Step 2: Solve for x.
- Step 3: Replace y with x to get f⁻¹(x).
10. How do you prove a function is bijective in exams?
To prove a function is bijective in exams, you must clearly prove injectivity and surjectivity separately.
- Injective proof: Assume f(a) = f(b) and show a = b.
- Surjective proof: Let y be in the codomain and solve f(x) = y.
- Conclude that the function is bijective if both are satisfied.
