Testing for symmetry about a curve/line In High School Algebra , after studying how to plot a graph of $f(x)$ (rather called $y$) against $x$ in Cartesian coordinates, we studied several tests to determine the symmetry of the plotted graphs about the $x$ and $y$ axes, and around the origin point.
The tests seemed pretty simple and intuitive, you can Google them if you need to.
However, when I  thought of investigating the symmetry of a certain function (say $f(x)$) about a given equation of a line or a curve (say $g(x)$), the solution did not seem to be that intuitive.
The case seemed trivial and easy when $g$ was simply $x=0$ or $y=0$, but what about other cases when $g(x)$ is in fact a function of $x$?
That is, How to investigate the symmetry of $f(x)$ around $g(x)$?
Another question that crossed my mind when thinking of the first one was the case when  $f(x)$ is not simply a single function, as in discussing whether two functions (say $v(x)$ and $s(x)$) formed a mirror image of each other around a third function (say $g(x)$) or not. The case again seems simple enough when $v(x) = inverse(s(x))$ , and $g(x)=x$. Although I know no test for that, but it is always the case that a function and its inverse are symmetric around $g(x)=x$. And, again, I can't think of a possible way to check for the symmetry of two separate curves about a third curve, or even generate a mirror image of  a given curve about another given curve (you may consider that last sentence a third part of the question).
 A: How do we define 'mirroring' or reflection across a line? If you have a line $\ell$ and a point $A$, the image of $A$ across $\ell$ is constructed by dropping a perpendicular from $A$ onto $\ell$, and doubling it (i.e. extending it by its own length) on the other side of $\ell$ to get a point $A'$ at the end of the segment. Reflection across a point $O$ is even simpler. To obtain the image of $A$, join $A$ and $O$ and double that segment, to get $A'$ at the end of the doubled segment.
We can define reflection across a circle, too: this is called inversion and it has many interesting properties. The same technique can be extended to a three-dimensional circle (a sphere), a four-dimensional circle, etc.
Note that all these 'reflections' have some common properties:


*

*Every point which has an image, has a unique image. (And with a little stretching of the concepts of geometry (eg. by introducing a point/line at infinity), you can even say that every point has an image.)

*If $A'$ is the image of $A$, then $A$ is the image of $A'$ in the same reflection.

*If a continuous function/locus/graph is reflected, the image is still continuous.


Suppose you wanted to define reflection across an arbitrary curve. To make things simpler, let's assume that this curve is actually a function of $x$ in the Cartesian plane. Obviously, a method of reflection like that for a point is not going to work. If we try to use the method for a line, we find that 'the perpendicular' from a point to an arbitrary curve is not unique -- in fact, whenever the curve is not a line (i.e. whenever the tangents to the curve at some two different points are not parallel), there will be at least one point with at least two perpendiculars of the same length. This violates all the three common properties.
Similarly, we cannot use the inversion method because that depends heavily on the [relatively nice] properties of a circle. In general, we do not know of a method of reflection that works for arbitrary curves and satisfies the three properties which, if you think about them, are quite reasonable expectations.
