In a math book I have to read, the definition of a function is shown similar to this: $$X:\Omega \to \mathbb{R}$$ $$e\mapsto X(e) \in \mathbb{R}$$ According to my understanding, the first line defines a function $X$, which takes one argument from the set $\Omega$ and produces some real number. Now the second line tells, when I take an $e$ and apply the function $X$, the result will be a real number.
Now, my question is: What is the purpose of the second line? Do I get any information, which was not already in the first line?
No, we don't get any new information from the second line.
However, if $X(e)$ in the second line were replaced with something like $100$ or $|e|$, then we will get new information (i.e. what the function actually outputs ).