Definition
A function is a relation that assigns exactly one output (value) to each input from a given set. Formally, a function $f$ from set $A$ to set $B$ is written as $f: A \to B$.
Types of Functions and Their Characteristics
- One-to-One (Injective):
- Each output is mapped by at most one input.
- If $f(a_1) = f(a_2) implies a_1 = a_2$.
- Onto (Surjective):
- Every element in the codomain is an output of the function.
- For every $b in B$, there exists $a in A$ such that $f(a) = b$.
- One-to-One Correspondence (Bijective):
- Both injective and surjective.
- Every element in the codomain is mapped by exactly one element in the domain.
- Constant Function:
- All inputs map to the same output.
- $f(x) = c$ for all $x$.
- Identity Function:
- Each input maps to itself.
- $f(x) = x$.
- Even and Odd Functions:
Worked Example
Let $f: mathbb{R} \to mathbb{R}$ be defined by $f(x) = 2x + 3$.
Suppose $f(a) = f(b)$:
$$
2a + 3 = 2b + 3 implies 2a = 2b implies a = b
$$
So, $f$ is injective.
For any $y in mathbb{R}$, solve $y = 2x + 3$:
$$
x = \frac{y - 3}{2}
$$
Since $x in mathbb{R}$ for all $y$, $f$ is surjective.
Since $f$ is both injective and surjective, it is bijective.