1. What an Identity Function Means
An identity function is a function that returns the input exactly as it is. Whatever value you give the function, the same value comes back as the output.
It acts like a perfect mirror: nothing changes when passed through the function.
2. Formal Definition
If A is any set, the identity function on A is defined by sending each element to itself.
2.1. Definition
I_A : A \to A
I_A(x) = x \quad \text{for every } x \in A
3. Examples of Identity Functions
These examples show how the identity function works in common cases.
3.1. Example 1
Identity function on real numbers:
I(x) = x
3.2. Example 2
Identity on a small set:
A = \{1,2,5\}
I_A = \{(1,1),(2,2),(5,5)\}
3.3. Example 3
Identity on letters:
A = \{a,b,c\}
I_A = \{(a,a),(b,b),(c,c)\}
4. Properties of the Identity Function
The identity function has several special features that make it important in many areas of mathematics.
4.1. 1. One-One
No two different inputs produce the same output — because the output is exactly the input.
4.2. 2. Onto
Every element in the set A appears as an output because each element maps to itself.
4.3. 3. Bijective
Since it is both one-one and onto, the identity function is always bijective.
4.4. 4. Acts as a Neutral Function
When composed with any function, the identity does not change anything.
f \circ I_A = f
I_B \circ f = f
4.5. 5. Inverse of Itself
The identity function is its own inverse:
I_A^{-1} = I_A
5. Why the Identity Function Matters
The identity function is the simplest possible function but plays an essential role in understanding more advanced ideas like composition, inverses, transformations, and structure of mappings. It serves as the baseline against which other functions can be compared.