1. What a Bijective Function Means
A bijective function is one that is both one-one (injective) and onto (surjective). This means:
- No two inputs share the same output (injective).
- Every element in the codomain is used as an output (surjective).
Because of this perfect pairing, bijective functions match each input with exactly one unique output, and every output has a matching input.
2. Formal Definition
A function f: A → B is bijective if it satisfies both injectivity and surjectivity.
2.1. Injective Condition
f(a_1) = f(a_2) \Rightarrow a_1 = a_2
2.2. Surjective Condition
\forall b \in B, \; \exists a \in A \text{ such that } f(a) = b
3. Why Bijective Functions Are Special
Bijective functions are important because they create a perfect one-to-one correspondence between the two sets. This allows the function to be reversed — meaning the inverse function exists and is well-defined.
4. Examples of Bijective Functions
Here are some functions that satisfy both one-one and onto conditions:
4.1. Example 1
f(x) = x + 3, with f: \mathbb{R} → \mathbb{R}
Every value in the codomain is hit by exactly one input, and different inputs give different outputs → bijective.
4.2. Example 2
f(x) = 2x - 5, domain and codomain both real numbers.
This function is strictly increasing, so it is one-one and covers all real values → bijective.
4.3. Example 3
Let A = {1,2,3}. Define:
f = \{(1,4),(2,5),(3,6)\}
Each input pairs uniquely with an output, and all outputs in the codomain appear → bijective.
5. Non-Examples (Not Bijective)
These functions fail either the one-one condition, the onto condition, or both.
5.1. Example 1 — Fails Injective
f(x) = x^2 on \( \mathbb{R} \)
Because f(2) = f(-2), the function is not one-one → not bijective.
5.2. Example 2 — Fails Surjective
f(x) = e^x
This never outputs zero or negative values, so it does not cover the entire codomain \( \mathbb{R} \) → not bijective.
6. Inverse Function Exists Only for Bijective Functions
Because bijective functions pair inputs and outputs perfectly, each output can be traced back to a unique input. This allows the inverse function \( f^{-1} \) to exist and be well-defined.
6.1. Example
If f(x) = 3x + 2, then:
f^{-1}(y) = \frac{y - 2}{3}
This works only because f is bijective — every output comes from exactly one input.
7. Why Bijective Functions Matter
Bijective functions are used to create correspondences between sets, build inverse functions, and study transformations where nothing is lost or repeated. They play a key role in algebra, geometry, and many higher-level ideas.