1. Why Functions Are Classified
Functions behave in many different ways, so grouping them into types makes it easier to understand their patterns. Some types focus on how inputs relate to outputs (like one-one or onto), while others focus on the function's formula or structure (like polynomial or rational).
2. Types Based on Input–Output Mapping
These types describe how the function connects inputs to outputs. They help understand whether the function is invertible or how tightly the mapping behaves.
2.1. One-One (Injective)
Different inputs always give different outputs. No two inputs share the same output.
2.2. Onto (Surjective)
Every element of the codomain appears as an output of the function.
2.3. Bijective
A function that is both one-one and onto. These functions have perfect pairing and always have inverses.
2.4. Constant
All inputs have the same output value.
2.5. Identity
Each input returns itself: f(x) = x.
3. Types Based on Formula or Expression
These types depend on the algebraic form of the function. They help when solving equations, graphing, or analysing behaviour.
3.1. Polynomial
Functions built from powers of x with coefficients, like x² + 3x + 1.
3.2. Rational
Functions written as a ratio of two polynomials.
3.3. Algebraic
Functions formed from algebraic operations: roots, powers, sums, products.
3.4. Real-Valued
Outputs are real numbers, even if inputs may be different types.
3.5. Vector-Valued
Outputs are vectors instead of single numbers.
4. Why These Classifications Help
Knowing the type of a function helps predict its graph, understand which rules apply, identify whether an inverse exists, and choose the right method for solving problems. This overview prepares the ground for studying each type in more detail.