Is a Circle Considered a Function?

[Loren Booda]

I was brought up believing that when a mapping had more than one ordinate (y) for a particular abscissa (x), it did not describe a function. So is a circle not a function?

 

[scarecrow]

correct.....

 

[dextercioby]

The so-called "vertical test" will always tell u the answer.In the case of the circle,there are an infinite number of portions/arches (is this the word?? ) which are functions.

Daniel.

 

[matt grime]

<pointing out what a function is> A function from where to where?

 

[Galileo]

There is some semantics involved.
Since a circle is a geometrical figure, the wording: 'is a circle a function' has no meaning.
For the same reason, a parabola is not a function, but the graph of the function is.

...doh, I `m getting old and picky. Nevermind...

 

[HallsofIvy]

Indeed, even if we interpret "is a parabola a function" to mean "is a relation whose graph is a parabola a function" we would have to specify the coordinate system. A parabola whose axis is parallel to the y-axis corresponds to a function but exactly the same parabola, in a rotated coordinate system would not be.

(Notice I switched from "circle" to "parabola". A circle, in any coordinate system, does not correspond to a function.)

 

[kishtik]

A circle, in any coordinate system, does not correspond to a function.

In any carthesian coordinate system.
In a polar coordinate system, it does.

 

[NateTG]

I mathematics, a function is a sort of black box - you put something in, and it spits something out. (There are other ways to think of them as well.) Now, you ask whether a circle is a function, and the answer is clearly no, since a circle is typically a set of points.

In the familiar (I suppose it could be called cartesian) system, there is no function that whose plot is a circle. In a polar or parametric system functions can readily have circular plots.

It is also relatively easy to see that:
$$f(x)=\sqrt{1-x}$$
generates the plot of a half circle - which can often be used instead.

 

[kishtik]

$$f(x)=\sqrt{1-x}$$
generates the plot of a half circle - which can often be used instead.

That is: $$f(x)=\sqrt{1-x^2}$$

 

[Galileo]

The plot of
$$f:[0,2\pi) \to \mathbb{R}^2$$
$$f(x)=(cos(x),sin(x))$$

is a circle.
Or is this cheating?

 

[matt grime]

This is the "non-cartesian function" people were mentioning. The question, from what i can tell, boils down to the fact that the OP thinks a graph and a function are the same thing.