Concept and expression of a real function

Concept of function

When we use the word it "depends" in everyday parlance we are indicating a dependence relation, for example when we say that the price of a call depends on its duration.

We call function $f$ that goes from the set $A$ to the set $B$ to a relation of dependence on which for any $x$ in set $A$ we assign only one element in set $B$.

It is represented using the following notation:$$\begin{array}{rcl}f:A& ⟶& B\\ \\ x& ⟶& y=f(x)\end{array}$$The set $A$ is called the domain, and the set $B$ the codomain.

If an element $x$ in set $A$ is assign to an element $y$ in $B$, it is said that $y$ is an image of $x$ under the function $f$, or that $x$ is an inverse image of $y$ under $f$.

If $A$ and $B$ are sets of real numbers, we speak about real function of real variables.

Analytical expression of a function

Sometimes a function can be expressed by means of a formula that allows to calculate the images of the elements of the domain and the inverse images of the elements of the codomain.

Let's consider for example the function $f:\mathbb{R}⟶\mathbb{R}$, where every real number $x$, is assigned to its double. We can represent that by $y=f(x)$ with: $f(x)=2x$ This formula is known as the analytical expression of the function $f$.

It is equivalent to writing $y=2x$.

In this case the variable $x$ receives the name of independent variable and the variable $y$ the name of dependent variable.