What do we mean by 'defining a function'?

Question

First I will start by quote from Wikipedia about function's defining methods :

There are many other ways of defining functions. Examples include piecewise definitions, induction or recursion, algebraic or analytic closure, limits, analytic continuation, infinite series, and as solutions to integral and differential equations. The lambda calculus provides a powerful and flexible syntax for defining and combining functions of several variables. In advanced mathematics, some functions exist because of an axiom, such as the Axiom of Choice.

I'm begin studying math at my own, I have started with Functions but I got confused regarding the term 'define a function' Is it really all those things 'induction,recursion,algebraic and analytic closure,limits, ... ' only for defining the Function ? then what do we mean by 'define the function' ? and why it's important so much that you can do it In many and diverse Ways ?

Note : I'm absolute beginner
The Math studying in the school ' also in the first years of university study' does not provide you The very basic Concepts like 'What really is function' or 'Derivation',Rather than you directly find yourself learning How to solve Function ! or How to do the 'Derivation' For me I do not like this approach, I love math but I can not continue in something I don't understand properly
Please Consider this Note while you help me in my issue

Answer 1

Fundamentally, defining a function $f:E\to F$ is to give a subset $A$ of $E\times F$ that verifies certain properties. Or in other words, to give a set $A$ of couples $(x,y) \in E\times F$, such that for every $x\in E $ you have one and only one $y \in F$ such that $(x,y) \in A$, and we often note this $y$ as $f(x)$.

This is how to define a function by the "fundamental definition of a function", but you can define a function by giving a set of properties that this function must verify. However, the hard part when you do it this way is to show that it indeed defines a function, and that this function is unique.

How can I define something that is not a function? Take this:

let $f:\mathbb{Z}\to \mathbb{R}$ the function that verify the following propriety:

$f(n)$ is the number $t \in \mathbb{R}$ such that $t^2 = n$

This defines a relation (not a function), as the set of couples $(n,y)$ that verify this property ("$y^2=n$") doesn't follow the axioms of a function:

• there is no $y \in \mathbb{R}$ such that $f(-1) = y$
• there is more than one $y \in \mathbb{R}$ such that $f(1) = y$

And the property that defines the function can be complex:

Let $f:\mathbb{R} \to \mathbb{R}$ be the function that is derivable and verify $\forall x\in\mathbb{R},\ f'(x) = f(x)$ and $f(0)=1$

And this indeed defines a function (even if it's not trivial to show that it defines a function, rather than a relation)

In the end, the limit is this: you can define a function by giving it a property that characterizes it completely

Answer 2

Intuitively, a function is just a way of assigning a (unique) output to every input in some set. For example, for any set, it's possible to define the identity function, whose output is the same as its input. Equivalently, we could write $f_1(x) = x$. Or if we are interested in numbers, we could have the function which doubles its input: $f_2(x) = 2x$.

Now already I've defined functions (described how to get an output from any valid input) in two different ways: in words, and through a formula. There's also the factorial function, which is given by multiplying all integers up to the input:

$$ n! = 1 \times 2 \times \ldots \times n $$

which can equivalently be defined "inductively" by

$$ 0! = 1 \\ n! = n \times (n-1)! $$

Normally, an author will choose to define a function in a way that makes its key properties easiest to prove; they might also use a definition that is most intuitive for some application, or do both and show that they are equivalent.

As a bit of personal advice, at this stage in your mathematical learning, don't worry too much about this in the abstract. It'll make sense as you encounter different functions in various contexts and you'll become used to all the ways that they can be defined.

And finally, to make what I said in the first paragraph more precise, and for the sake of completeness, normally a function $f$ with domain $A$ and range $B$ is defined as a subset of the Cartesian product $A \times B$ such that for every $a_0 \in A$, there is a unique pair $(a, b) \in f$ with $a_0 = a$, which we notate as $f(a_0) = b$.