Definition:Real Function

Definition
A real function is a mapping or function whose domain and codomain are subsets of the set of real numbers $\R$.

It is frequently understood in many areas of mathematics that the domain and codomain of any function under discussion are of the set of real numbers, so the adjective real is usually omitted unless it is an important point to stress.

Because the concept of a function has been around for a lot longer than that of a general mapping, there is a lot more terminology that has developed up round the subject.

Formula
A function $f: S \to T$ can be considered as a formula (or sometimes as a rule) which tells us how to determine what the value of $y \in T$ is when we have selected a value for $x \in S$.

As an Equation
It is often convenient to refer to an equation or formula as though it were a function.

What is meant is that the equation defines the function, i.e. it specifies the rule by which we obtain the value of $y$ from a given $x$.

For example, let $x, y \in \R$.

Let $f: \R \to \R$ be defined by $\forall x \in \R: f \left({x}\right) = x^2$.

We may express this as $y = x^2$, and use this equation to define this function.

This may be conceived as:
 * For each $x \in \R$, the number $y \in \R$ assigned to it is that which we get by squaring $x$.

Function of Two Variables
Let $f: S \times T \to \R$ be a mapping where $S, T \subseteq \R$.

Then $f$ is defined as a (real) function of two (independent) variables.

The expression:


 * $z = f \left({x, y}\right)$

means:
 * (The dependent variable) $z$ is a function of (the independent variables) $x$ and $y$.

Function of n Variables
The concept can be extended to as many independent variables as required.

Let $f: S_1 \times S_2 \times \cdots \times S_n \to \R$ be a mapping where $S_1, S_2, \ldots, S_n \subseteq \R$.

Then $f$ is defined as a (real) function of $n$ (independent) variables.

The expression:
 * $y = f \left({x_1, x_2, \ldots, x_n}\right)$

means:
 * (The dependent variable) $y$ is a function of (the independent variables) $x_1, x_2, \ldots, x_n$.

Also see

 * Real-Valued function