> Start from deep points TOP> 3. What are functions?
3. What are functions?
THAT IS,
if we input something in it,
it shows one solution
and we call such system as a function.
A vending machine is also one kind of functions as its system,
because if we input money in it,
it gives us only one canned juice at once.
If it gave us two cans at the same time,
that situation means that it gives us two solutions.
So we do not call its system a function.
(In ordinary, we call such situation as "out of order.")
One of the simplest examples is old ticket vender machine systems of trains.
If we input 1 dollar in machines with a sign, 1 dollar tickets,
it gives us only one 1 dollar ticket.
This action is same at anytime for everyone.
Its action is very important,
because if it gave us different tickets for same amount money,
that is, if it gave different solutions to same matter of input
we never call such a whole system as a function.
This situation means that
every time that system itself changes.
So we distinguish those systems as different functions.
In math cases, for example,
we have a formula,
y=4x-3.
It is a relational expression,
such as if we change x's value, y's value also changes corresponding to x.
And how to decide y's value depends on a system,
that is, in this case,
"to 4 times of x minus 3, then an ejected answer is y."
In this equation,
if we input some number into x (we exchange x into that number alternatively.),
it gives us the same number as an answer,
which is the same to every same input number.
If it gives us two and more answers,
we do not call it as a function.
If we input different values into x,
those answers are same in accident.
That situation is acceptable for functions.
If I express functions in graphs for example,
function graphs are as two below examples,
one x's value corresponds to one y's value.
Not function graphs are as two below examples,
two and more values of y correspond to one x's value.
I will give you an example that same result follows from different inputs.
This is also a function.
In this case, two different values of input x turns into the same value of y accidentally.
In this case, four different values of input x turns into the same value of y accidentally.
Different and various shaped graphs exists in this world.
