1. What a One-One Function Means
A one-one (or injective) function is one where different inputs always give different outputs. No two inputs share the same output value.
In simple terms: the function never “collides” two inputs into one output.
2. Formal Definition
A function f: A → B is one-one if every pair of different inputs has different outputs.
2.1. Definition
f(a_1) = f(a_2) \Rightarrow a_1 = a_2
\text{or equivalently, } a_1 \ne a_2 \Rightarrow f(a_1) \ne f(a_2)
3. How to Check if a Function Is One-One
You can check injectivity in different ways depending on how the function is given.
3.1. 1. Using Algebra
Set f(a1) = f(a2) and simplify. If it forces a1 = a2, the function is one-one.
3.2. 2. Using Graph
A function is one-one if any horizontal line intersects the graph at most once. (Horizontal Line Test)
3.3. 3. Using Ordered Pairs
Check if any two different inputs share the same output. If not, it is injective.
4. Examples of One-One Functions
Here are some functions that satisfy the one-one condition:
4.1. Example 1
f(x) = 2x + 3
If 2a + 3 = 2b + 3, then a = b. So the function is one-one.
4.2. Example 2
f(x) = x^3
Cubic functions strictly increase or decrease, so they never repeat an output.
4.3. Example 3
From ordered pairs:
f = \{(1,4),(2,6),(3,8)\}
Each input has a unique output → one-one.
5. Non-Examples (Not One-One)
These functions repeat output values, so they are not injective.
5.1. Example 1
f(x) = x^2
f(2) = 4 and f(-2) = 4 → different inputs give same output.
5.2. Example 2
From ordered pairs:
g = \{(1,5),(2,5),(3,7)\}
Both 1 and 2 map to 5 → not one-one.
6. Why One-One Functions Matter
One-one functions never merge information, so they preserve distinctness. This makes them important in building inverse functions and in transformations where every input must be traceable back from its output.