Why we write $y=f(x)$ instead of $f(x)$? For a function I know we can write
\begin{align}
f: &\, \mathbb{R} \rightarrow \mathbb R \\
&x\mapsto f(x)
\end{align}
However, I have also seen 
\begin{align}
f: &\, \mathbb{R} \rightarrow \mathbb R \\
&x\mapsto y=f(x)
\end{align}
I don't grasp the meaning of $y$, what does $y$ mean?
Why write $y=f(x)$ instead of $f(x)$?
Thanks!
 A: The convention is essentially useless (but not harmful) if domain and codomain are the same.
Suppose domain and codomain are different, for instance
\begin{align}
f: &\, \mathbb{R} \rightarrow \mathbb R^2 \\
&x\mapsto f(x)
\end{align}
The convention of writing
\begin{align}
f: &\, \mathbb{R} \rightarrow \mathbb R^2 \\
&x\mapsto y=f(x)
\end{align}
allows us to distinguish between elements of the domain, which we shall refer to as $x$, and elements of the codomai, which we shall refer to as $y$. Basically, as soon as you see $y$ you should be thinking of a vector in $\mathbb R^2$, while if you see $x$ you should be thinking of a real number. This makes immediate that expressions
$$y\cdot x$$
$$x^2+y$$
are not well defined, while 
$$y\cdot y$$
$$x+x^2$$
are perfectly fine.
A: You can, simply because you are working in a 2D space, with two axis : $x$ and $y$. The representative curve of your function is made up of all the points $A(x;f(x))$ so this curve has the equation $y = f(x)$
A: 'y' is just a variable, if you write y = f(x) it means value of y  depends on the value of x i.e., for different values of x , there is a function f(x) which computes the value of y.
hence x is called as "independent" variable and y is known as "dependent" variable on x
it's a formal way of representing that y is dependent on the value of x and x is independent
A: $$x\mapsto f(x)$$ a map that takes $x$ to an element, following the rule represented by $f(x)$.
$$x\mapsto y=f(x)$$ a map that takes $x$ to an element (let's call it $y$), following the rule represented by $f(x)$.
