What is the name for relations like sideways parabolas that are not functions?

[Thadriel]

All I've ever seen is how to determine if something is a function or not. A vertical parabola is a function. But what set of relations do sideways parabolas fall in? To be precise, what do we call a relation that has two y outputs for every one x input, rather than being one-to-one like a function? Is there a term, or do we just say, "Not-a-function?" Is it just called a "one-to-many" relation?

I don't recall ever going over this in any math class I took. If we did, it was probably five minutes. So, thanks for any info on the term to describe things like [imath]y^2 = x[/imath].

 

[lev888]

All I've ever seen is how to determine if something is a function or not. A vertical parabola is a function. But what set of relations do sideways parabolas fall in? To be precise, what do we call a relation that has two y outputs for every one x input, rather than being one-to-one like a function? Is there a term, or do we just say, "Not-a-function?" Is it just called a "one-to-many" relation?

I don't recall ever going over this in any math class I took. If we did, it was probably five minutes. So, thanks for any info on the term to describe things like [imath]y^2 = x[/imath].

We can still call x=y² a function. It's a convention to have a horizontal independent variable axis and a vertical function value axis. Doesn't mean other arrangements are prohibited. As long as you describe what's going on, I think it's fine to swap the axes to define a "horizontal" parabola as a function.

 

[topsquark]

All I've ever seen is how to determine if something is a function or not. A vertical parabola is a function. But what set of relations do sideways parabolas fall in? To be precise, what do we call a relation that has two y outputs for every one x input, rather than being one-to-one like a function? Is there a term, or do we just say, "Not-a-function?" Is it just called a "one-to-many" relation?

I don't recall ever going over this in any math class I took. If we did, it was probably five minutes. So, thanks for any info on the term to describe things like [imath]y^2 = x[/imath].

Any mapping defined by a set of (x, y) pairs is called a "relation." If a relation passes the vertical line test it is called a "function." So [math]y^2 = x[/math] is not a function y(x) but a relation. (It's still called a parabola, though.) However as lev888 points out x(y) is a function in this case.

Is that what you are looking for?

-Dan

 

[Otis]

do we just say, "Not-a-function?

Hi. I would say that y^2 is not a function of x. If we treat each half of the parabola separately, then those are each functions of x.

y_top = sqrt(x)

y_btm = -sqrt(x)

Do you remember a topic called conic sections? (Click the 'Parabola' caption). It's another way to define parabolas in the xy-plane.

y^2 = 4ax

where a is half the distance between the parabola's focus and its directrix.

?

[imath]\;[/imath]

 

[Thadriel]

Any mapping defined by a set of (x, y) pairs is called a "relation." If a relation passes the vertical line test it is called a "function." So [math]y^2 = x[/math] is not a function y(x) but a relation. (It's still called a parabola, though.) However as lev888 points out x(y) is a function in this case.

Is that what you are looking for?

-Dan

I was hoping for a special word with a similar use to "function" to describe this particular type of relation (a word of a similar categorical hierarchy level), but I suppose if there isn't one, there isn't one. My example wasn't the best one. I think I have a better one below.

Hi. I would say that y^2 is not a function of x. If we treat each half of the parabola separately, then those are each functions of x.

y_top = sqrt(x)

y_btm = -sqrt(x)

Do you remember a topic called conic sections? (Click the 'Parabola' caption). It's another way to define parabolas in the xy-plane.

y^2 = 4ax

where a is half the distance between the parabola's focus and its directrix.

?

[imath]\;[/imath]

Sure, but if you graph that you need two separate graphs, as you pointed out, and the domains of either one do not include negative numbers; but in x(y) = y^2, the domain goes across all real numbers. It seems to be pedantic though, since the combined domains do include all real numbers.


But anyway, that's a side discussion. I was just wondering if we would have a special name for a relation that has two outputs for a single input. That was just the first example that came into my head, and it's obviously not a useful one. What would we call this then?

Is there a special word to describe this?

.
.
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I'm assuming the best I could hope for is a "one-to-many" relation. It would make sense for there not to be a particular word to describe them, given that functions are so useful.



Thanks for your insight everyone!

 

[BigBeachBanana]

Hi Thadriel,
I believe this is what you're looking for.

Injective, Surjective and Bijective