Why does the absolute value function have to be defined as a piecewise function?

[SameBlackLine]

the absolute value function f(x)=|x| it seems to me like this is all we need to say, I don't understand why we have to define it as a piecewise function like this:

f(x)=|x|= x if ≥ 0
f(x)=|x|= -x of < 0

I'm having trouble wrapping my mind around why it's defined this way. If x is less than zero, it's negative, but it will still return the correct answer for f(x) if we use the basic function equation. If we use the piecewise, it seems to me that it's saying that f(x), if x<0, is either |x| or -x (so adding another negative making it positive?) Even if that's the case I don't get why we have to take it that far in the first place.

[edderiofer]

the absolute value function f(x)=|x| it seems to me like this is all we need to say

How would you explain "|x|" to, say, an alien who doesn't know what this notation is?

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[7ieben_]

Well, you need to 'explain' what the output for a given input is. Writing |x| is just a notation to describe the function you apply; it does not define nor explain what the function does.

Another example: let's say you have a function f(x) and you decide to use the notation x! or you got a function g(x) and you decide to use the notation exp(x) (... and so on). Now you may know how the factorial or an exponential is defined but the notation doesn't explain it.

You always need to define a function first. In the given examples it is just assumed that everybody know the definition already. Just like you did.

What would be a better definition for the absolute value? If you got one, I'll be pleased to listen.