Functions — Definition, Domain and Range

A function is a rule that assigns to every element of one set (the domain) exactly one element of another set (the codomain). In business mathematics we almost always work with real-valued functions of a real variable, written $f: \mathbb{R} \to \mathbb{R}$ or simply $y = f(x)$.

If $x$ is the input (the independent variable) and $y = f(x)$ is the output (the dependent variable), then:

• The domain of $f$ is the set of all permissible values of $x$ for which $f(x)$ is defined.
• The range of $f$ is the set of all values that $f(x)$ actually takes as $x$ varies over the domain.

For a function used in business contexts — a cost function, a revenue function, a demand function — the domain is often further restricted by the real-world meaning of the variable (for instance, a quantity produced cannot be negative), even where the algebraic expression would be defined on a larger set.

A quick way to decide whether a value must be excluded from the domain is to look for two things: (a) a denominator that could become zero, and (b) an expression under a square root (or any even root) that could become negative. Both make $f(x)$ undefined at that $x$.