Definition:Real Function

A real function is a mapping or function whose domain and range are subsets of the set of real numbers $$\mathbb{R}$$.

It is frequently understood in many areas of mathematics that the domain and range 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.

Independent Variable
Let $$f: \mathbb{R} \to \mathbb{R}$$ be a (real) function.

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

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

Dependent Variable
Let $$f: \mathbb{R} \to \mathbb{R}$$ be a (real) function.

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

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

Domain
The domain of a (real) function needs to be understood. However, if it is not explicitly specified (as is frequently the case), then it is understood to consist of all the values in $$\mathbb{R}$$ 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 (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 $$x, y \in \mathbb{R}$$.

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

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

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

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

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

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) $$x_1, x_2, \ldots, x_n$$."

Comment
The terminology of "independent variable" and "dependent variable" arise from the concept that it is usual to be able to consider that $$x$$ can be chosen "independently" of $$y$$, but having chosen $$x$$, the value of $$y$$ then "depends" on the value of $$x$$.