## 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:$$$ \begin{array}{rcl}f: A &\longrightarrow &B \\\\ x &\longrightarrow &y=f(x) \end{array}$$$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 $$f: \mathbb{R} \longrightarrow \mathbb{R}$$, 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.

Write the analytical expression of the function $$f$$ that assigns to every real number the triple of its square minus one.

A real number is $$x$$. The square of $$x$$ is: $$$x^2$$$

The triple of the square of $$x$$ is: $$$3x^2$$$ The triple of the square of $$x$$ minus one is: $$$3x^2-1$$$

And therefore we have:$$$f(x)=3x^2-1$$$