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I guess i might be answering my own question here, but is it because of how we chose to represent our functions? That is, for the lack of better term, strech it along the x axis? Has it it been f(y) would it be the other way around?

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A function from some set $X$ to $\mathbb R$ is, by definition, a specific type of relation, which is a collection of pairs, $f\subseteq \{(x, y)\in X\times \mathbb R\}$. $f$ is a function if each $x\in X$ appears in precisely one pair in $f$, which is why you can write $f(x) = y$ instead of $(x,y)\in f$.

The reason why we are interested in functions is not really mathematical (mathematitians don't mind analyzing other types of relations) but more "real-life" in that very very often, functions are a very good way of describing things in the world. For example, if you look at how some object moves through time, then at each point in time, it is on only one location. Therefore, it is easy to see that you can see the position of the object as a function of time.

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By definition, f(x) is a manipulation on the number x such that we get another number, y.

Since any function must be clearly defined, when we input a value of x we should get a unique number, not a set of numbers. Therefore, one value of x must only return one value of y. On the other hand, nothing is stopping a specific value of y to return multiple x values, and is commonplace in periodic functions, like trigonometric functions.

So in a way, you did answer your own question. This property is due to the limits of our definition, not a mathematical phenomenon.

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The definition of a mapping states that each anticident in the initial set i.e each $x$ shall have one and only one image in the final set moreover this mapping can be a non injective mapping thus there exist maybe more than 1 $image$ $y$ having more than one anticident $x$ ! Note that the anticidents are different as stated in the beggining so you can call the relation $f$ a mapping. Thats why every $x$ in a mapping is related to only 1 image while there may exist Images that are related to several different Anticidents.

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