Math question--hello, fivemack

[redbird]

This is, I think, more technicality/terminology than anything. One of my fellow editors ran into this:

Is y-0 [equivalently, f(x)=0] a function that is symmetric with respect to the x-axis?

It seems to me and her that it is, but a site she often finds reliable claims that there are no such functions.

[Note: this is a technical point, not that she and I have forgotten how to graph simple curves: y=x² is a function, but x=y² is not a function but a relation.]

ETA: Thank you all. Within a couple of hours, I was able to go back to Marta and tell her that I'd had responses from people I trusted, including a Ph.D. mathematician, and she and I were right and the web site was wrong. When she first asked me, after lunch, she wasn't going to consult her usual source because the manuscript is due Monday.

Comments

[drplokta]

Yes, f(x)=0 is a function and is symmetric with respect to the x axis. A claim that there are no such functions is equivalent to a claim that real numbers have two square roots -- it's broadly true, but ignores the degenerate case.

[drplokta]

Oh, and technically x=y² is a function on y -- you've merely reversed the traditional nomenclature and you're using x to refer to f(y) instead of using y to refer to f(x). What's not a function is f(x)=sqrt(x).

[amaebi.livejournal.com]

In case more backup is useful: Yes, f(x) = y = 0 is a function of x, and it's also symmetric around the x axis.

[fivemack.livejournal.com]

Yes, f(x)=0 is the only function that is symmetric with respect to the X-axis, the requirement being f(x) = -f(x) (rather than f(x) = f(-x), which is symmetry with respect to the Y-axis, or f(x) = -f(-x), which is probably symmetry with respect to the line y=-x but which I'd just call an odd function)

[aquaeri.livejournal.com]

Another vote for "of course f(x)=0 is a function". The "reliable" site probably forgot about the degenerate case.

Definitions....?

[dragon3.livejournal.com]

If I recall correctly a function of x *must* have a *single* value for all values of x (except at singular points where the limit depends on the direction of approach -- think tan(pi/2))

If symmetry requires that every point have a corresponding point eqi-distant from the x axis in the opposite direction, then y=0 is not symmetric, as there is only a single point corresponding to each x value.

I'm not sure that there is a valuable answer. Certainly not without a rigorous definition of both function and symmetry. If anybody is really keen -- seek out Joukowski Transforms that will translate a circle into a flat line of finite length where there really are two y values for each x value, but both of them are zero, except where there is no y value ;-)