Functions, domain and range

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.

In this mapping, the outputs attained are \(y_1,y_2,y_3,y_4\); \(y_5\) lies in the codomain but not the range.

Function notation names the input: \(\mathrm{C}(n)=4n+7\) can model the cost of \(n\) items. If at most \(20\) items can be ordered, the practical domain is \(n\in\{0,1,\ldots,20\}\), not every real number.

Without a stated restriction, use the largest real domain where the formula is defined. Denominators cannot be zero, logarithm arguments must be positive and real square-root arguments must be non-negative.

Example

For \(\mathrm{f}(x)=\sqrt{x-3}\), the domain is \([3,\infty)\) and range is \([0,\infty)\).

Range on a restricted domain

Treat the stated domain as part of the function. First sketch only the permitted inputs. Then project the remaining graph onto the vertical axis. An excluded input does not automatically exclude its output: another permitted input may give the same value.

For a quadratic, compare the endpoint heights and the vertex height, but use the vertex only if its input lies in the domain. With an unbounded domain also check the end behaviour. A bracket is closed precisely when some permitted input attains that output.

For \(\mathrm f(x)=(x-1)^2\) on \(0<x\le3\), the minimum \(0\) is attained at \(x=1\), and the maximum \(4\) at \(x=3\); the range is \([0,4]\). The excluded input \(0\) does not remove the output \(1\), since \(\mathrm f(2)=1\).

For a monotonic function on an interval, trace the two endpoint values, reversing their order if the function decreases. If a denominator vanishes inside the interval, split the domain there before finding the range; never join the graph across an excluded input.