Types of Functions
Types of Functions in Mathematics
Definition: A function is a relation where each input has exactly one output.
1. Based on Mapping
| Type | Description | Example |
|---|
| One-One (Injective) | Each input maps to a unique output | f(x) = 2x + 1 |
| Onto (Surjective) | Every output has a pre-image in the domain | f(x) = x³ (if domain = codomain = ℝ) |
| Bijective | Both injective and surjective | f(x) = x + 5 |
| Many-One | Multiple inputs have the same output | f(x) = x² |
2. Based on Expressions
| Type | Description | Example |
|---|
| Algebraic Function | Uses algebraic expressions | f(x) = 3x² + 2x + 1 |
| Polynomial Function | Polynomial of x | f(x) = x³ - 2x + 5 |
| Rational Function | Ratio of two polynomials | f(x) = (x² + 1)/(x - 3) |
| Irrational Function | Involves roots | f(x) = √(x + 4) |
3. Based on Values
| Type | Description | Example |
|---|
| Constant Function | Always gives same output | f(x) = 7 |
| Identity Function | Output equals input | f(x) = x |
| Modulus Function | Always gives non-negative value | f(x) = |x| |
| Signum Function | Gives sign of the number |
f(x) = {
1 if x > 0
0 if x = 0
-1 if x < 0
}
|
4. Special Types
| Type | Description | Example |
|---|
| Step Function (Greatest Integer) | Largest integer ≤ x | f(x) = ⌊x⌋ |
| Exponential Function | Variable is in exponent | f(x) = 2^x |
| Logarithmic Function | Inverse of exponential | f(x) = log(x) |
| Trigonometric Function | Based on angles | f(x) = sin x, cos x |
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