Functions and Relationships

A function is a relation between a set of inputs and a set of permissible outputs such that each input is related to exactly one output.

The domain is the set of permitted inputs. The codomain is the permissible target set, while the range is the set of outputs actually attained; hence the range is a subset of the codomain.

One-one relationship

A one-one function maps distinct inputs to distinct outputs. On its attained range, the inverse relation is also a function.

One-many relationship

A one-many relation associates one input with more than one output, so it is not a function. Its inverse relation may or may not be a function.

Many-one relationship

A many-one relation may be a function, but its inverse relation is not a function because one inverse input would have several outputs.

Vertical line testing

For every input in the declared domain, the vertical line must meet the graph exactly once. A vertical line may meet the graph at most once overall. To disprove the function claim, show an input in the declared domain with no output or with more than one output.

Horizontal line testing

A function is one-one if every horizontal line cuts its graph at at most one point. To show it is not one-one, give a specific horizontal line \(y=k\) that cuts the graph twice or more.

A piecewise function is defined by several sub-functions, with each rule applying on a stated interval or subset of the input domain.

The interval conditions are part of the definition: they determine which rule is used and whether boundary points are included or excluded.

For a function defined on all real inputs, a positive number \(p\) is a period when

\[\mathrm f(x+p)=\mathrm f(x)\qquad\text{for every}\hspace{0.5em}x\in\mathbb R.\]

Its graph therefore repeats after every \(p\) units along the horizontal axis.