Confusion with functions: Inputs and outputs

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Megannnnn

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Hello, everyone.
I need help with this, to understand what is going on in it.
I don't understand why a number is being assigned to the other number specifically, meaning for example: why is f(1)=2, and f(2)=3? Why can't f(1)=1 or f(2)=2?
(the same for the 'g' functions below it)
My question is if someone can please explain to me the reasoning behind the resulting outputs from the functions and what rules are guiding how it works.
Thank you very much!
 

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Functions should have very clear definitions. They don't just magically spit out numbers.

If f(x) = x+1, then f(1) = 2 and f(2) = 3. f(1) CANNOT be 1, because that is contrary to the definition of f(x).
 
Megannnnn said:
Hello, everyone.
I need help with this, to understand what is going on in it.
I don't understand why a number is being assigned to the other number specifically, meaning for example: why is f(1)=2, and f(2)=3? Why can't f(1)=1 or f(2)=2?
(the same for the 'g' functions below it)
My question is if someone can please explain to me the reasoning behind the resulting outputs from the functions and what rules are guiding how it works.
Thank you very much!
Click to expand...
This is just an example; they arbitrarily defined the function f by telling you that f(1) = 2 and f(2) = 3 (stating the value for each element of the domain). They could have picked any function at all. but this is the one they chose.

It sounds like you don't fully understand the concept of function yet (which is not uncommon). A function doesn't need a "rule guiding how it works" (though often there is, in the form of an equation). It is simply any association of outputs to inputs. A function is nothing more or less than a set of choices, which may or may not have a reason. (Part of the reason for inventing the concept of function was to separate it from the idea of equations, which got too hard to define once mathematicians tried to explain what calculus did: it changes one function to another, and they can't always be written as equations.)
 
I want to disagree, just slightly, but the more I think on it, I actually don't disagree at all, so maybe just some expansion.

If you are given f(1) = 2 and f(2) = 3, then that is what you have. That is the function's definition. Does it say anything about f(3)? No. But what if you REALLY want it to say something about f(3)? Nope. More information is needed. I invented a function that reproduces the given values, f(x) = x+1, but there are infinitely many of those, so this is not a particularly great accomplishment.

We might think a choice is arbitrary, but is it? It may be that there is a very nice and clean definition, but we just aren't told what it is. No matter. We can do many things with this - just knowing there IS a function. Knowing the exact definition is not always a great way to start a research project. The salient features of the project may be in the search itself.

A function needs a definition. Such a definition can come in many forms.

Instructive Example:
Q: What is the next number in this sequence? 1, 2, 3, 4, 5, 6
A: -2.
Why? How? It doesn't matter. I made up the sequence. Unless I'm trying to model some real phenomenon, or repeat someone experiment, I can define whatever I want.

If we stick with the definition given, assuming only the index numbers in the order presented: f(1) = 1, f(2) = 2, f(3) = 3, f(4) = 4, f(5) = 5, f(6) = 6, we know nothing precise about f(7). What does it tell us about f(4.7)? Nothing. f(0)? Nothing. We MIGHT be able to construe that it is a linear function. That would be helpful. But MUST we do that? No.

A function needs a clear definition if we need to know that for whatever it is that we are doing. Some functions lead to more information about surrounding values. Some don't.

Anyway, some rambling for the day.
 
It says “for example.” That is all it is, an arbitrary example.
 
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