The everyday meaning of the word "function"
and the mathematical meaning of the word "function" are similar, but not quite the same.
And the difference is important. In everyday use, one might say something like:
"how tired you are is a function of how much you slept the night before".
And so this indicates that there is a relationship between two quantities:
how much you slept and how tired you are.
And that knowing how much you slept will tell you something about how tired you are.
But it wouldn't tell you everything.
Merely knowing how much you slept the night before would not be enough to determine uniquely,
without any uncertainty, how tired you are.
Because how tired you are might depend on other things: how much you've eaten, how much coffee you've had,
how often your cat hit you on the head last night when you were trying to sleep.
So, again, knowing how much you slept the night before does not uniquely determine how tired you are.
But when you're talking about mathematical functions, the relationship is unique.
By this I mean that in order to figure out the value of the output, in this context that would be how tired you are,
you only need to know the value of the input. The output depends only on the input, not on anything else.
In math, a function is deterministic; that means that the output is determined by the input.
The output depends entirely on the input. So, as an example, lets go back to the square plus one function.
So, there's our rule: give the function an input x, square it and add one, and that's the output.
So, I don't know, we saw last time that the g of three is three squared plus one, equals ten.
And I can write it this way also: g, there is three coming in, and three has g acting on it
and the output is ten. So now, suppose I ask again what's g of three. And the answer - well, we've already done it - it's ten. OK.
It seems silly to do this twice, but this underscores, this highlights what this determinism,
what this deterministic property means.
It says that if you tell me the input, the output is determined.
So the same input gives the same output everytime.
The function never makes a mistake, there's no element of chance, same input always gives you the same output.
For this function, a little while later g of three is still ten, and a little while later still, g of three is still ten.
So we'll be studying deterministic functions in this course.
Deterministic in the sense: same input gives you the same output; all you need to determine the output is the value of the input.
So that's determinism, in this context.