A real function may be defined thus:
A real function of one variable is a set $f$ of ordered pairs of real numbers such that for every real number $a$ (from the domain of $f$) there is exactly one real number $b$ for which the ordered pair $(a,b)$ is a member of $f$. In this case we say that $f(a)$ is defined and we write $f(a)=b$. The number $b$ is called the value of $f$ at $a$. While plotting the graph of a function we usually write $y = f(a).$
I am confused here with the vocabulary we use while defining functions. If $f$ is a set, then what is meant by $f$ having a value at $a$? Do sets have something called 'values'?
We also say that, for example, the position $y$ or $x(t)$ of some particle is a function of time, i.e. $y=x(t)$. Does it mean that $y$ is a set of ordered pairs? But, the position is thought of as a variable.
We say that the derivative of a function - let us say $f$ - with respect to some variable $x$ is the rate of change of that function with respect to $x$. Now, if a function is a set, then what is meant by the rate of change of 'sets'?
It seems to me that a function $f$ is a variable which or whose values depend upon or controlled by some other variable $x$ called the independent variable.