- #26

pbuk

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For which bit? You could take Mathworld as an authoritative external reference.We need an umpire perhaps.

No, that relation is not a complete description of a function; in order to make it complete we need to specify theOk. So y = sin(x) is a function but

*domain*of the function, which is

**the**set of values that contains exactly all values that ## x ## is allowed to take, and the

*range*of the function, which is

**a**set of values that contains at least every corresponding value of ## y ##.

It might be: if we specify the domain ## y \in [0, 1] ## and the range ## x \in [0, \frac{\pi}2] ## then we have a function but there is no such function with the range ## x \in [0, \pi] ##.X=sin-1(y) is not.

I'm not sure what this bit means.Thete will be ‘functions’ which are functions in a limited range. Difficult but my insistence on some sort of qualification needing to be added to any function is justified.

Note that a function can also be many-to-one (but not one-to-many or many-to-many).Anything as sweeping as the one-to-one condition has to be qualified with limits.

Yes, mathematicians often do - but only when it matters. Like most scientists mathematicians often omit a lot of detail and assumed prior knowledge, if we didn't then mathematical texts would be unreadable. OK, even more unreadable.Do people ever actually do that?

I probably ought to say that there is also something called a 'many valued function', of which ## y = \sin^{-1}x, x \in \mathbb R, y \in [-1, 1] ## is an example but this is not a special kind of function, it is a completely different thing ('many-valued' is not an adjective here, rather the whole expression is a compound noun).

Perhaps - it is only 40 years since I learned this, almost to the day. I wouldn't be surprised though if the single-valued criterion goes back to the 19th century.Perhaps the vocabulary has changed in 50 years.