Function Key concepts
A Function is a precise assignment: every input in the Domain gets exactly one output. This “exactly one” condition is what separates functions from more general relations.
Concepts to keep separate
- Domain: the allowed input set.
- Codomain: the declared target set; see Codomain and range.
- Range/image: the outputs actually reached.
- Rule/formula: one way of describing the assignment, but not the whole function.
- Graph: a visual representation, usually a subset of a Cartesian product.
Mapping properties
- Injective: different inputs never share the same output.
- Surjective: every element of the codomain is hit.
- Bijective: both injective and surjective, so a genuine inverse exists.
Operations
Functions can be added, multiplied, composed, restricted to smaller domains, extended to larger domains, or inverted when conditions allow. In calculus, the main operations are differentiation and integration; in physics, the same functions often represent measurable quantities such as position, Velocity, field strength, or Energy.
Good habit: whenever a function appears, write before manipulating it.
Practical check
Before using algebra or calculus, decide whether the function is being studied symbolically, numerically, geometrically, or physically. The same function can be represented by a formula, table, graph, code routine, or measurement process, but each representation hides different assumptions.