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.
Best Answer
All of the terminology isn't meant to apply to sets in general. It only applies to particular types of sets the author uses to define functions.
If $f$ is a function as he defines it, then what is meant by "$f$ having a value at $a$" is that there is indeed an ordered pair "in the function" which has $a$ as its first element, and the second element of that ordered pair is the value of the function at $a$. The fact that this is a function guarantees that there is at most one such ordered pair so the function's value at $a$ is not ambiguous. He says this explicitly.
I think you are perhaps focusing too much on the ordered pair aspect. It can be discussed in an extremely rigorous fashion, but it is also customary to relax the rigor and write (rather sloppily, perhaps) that $b=f(a)$ instead of $(a,b)\in f$. The point is probably for you to be able to manipulate these symbols in a mathematically concise way, but to eventually realize that all of that can go on "in the background" without having to be written down every single time.