Definition:Real Function

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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 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.


Independent Variable

Let $f: \R \to \R$ be a real function.

Let $f \left({x}\right) = y$.

Then $x$ is referred to as an independent variable.


Dependent Variable

Let $f: \R \to \R$ be a (real) function.

Let $f \left({x}\right) = y$.

Then $y$ is referred to as a dependent variable.


Domain

Let $f: S \to \R$ be a real function.

The domain of $f$ is the set $S$.

It is frequently the case that $S$ is not explicitly specified. If this is so, then it is understood that the domain is to consist of all the values in $\R$ for which the function is defined.

This often needs to be determined as a separate exercise in itself, by investigating the nature of the function in question.


Formula

A function $f: S \to T$ can be considered as a formula 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; that is, it specifies the rule by which we obtain the value of $y$ from a given $x$.


Square Function

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


The (real) square function is the real function $f: \R \to \R$ defined as:

$\forall x \in \R: \map f x = x^2$


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 $x$.


Another useful notation is:

$\forall x \in \R: x \mapsto x^2$


Examples

Square Function

The (real) square function is the real function $f: \R \to \R$ defined as:

$\forall x \in \R: \map f x = x^2$


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

  • Results about Real Functions can be found here.


Sources