What would a definiton of reflection symmetry be?

Answer 1

There are several types of reflection symmetry, the most common ones considered in HS or undergrad college algebra are symmetry wrt the origin, wrt the y axis, and wrt the x axis.
Symetric wrt the origin
f(-x) = -f(x) for all x in domain example y = x^3
symmetric wrt the y axis
f(x) = f(-x) for all x in the domain example y = x^2
symetric with respect to the x axis
this would not be a function in the usual sense, but each x would generate two y's, which are equal in absolute value and opposite in sign.
example x^2 + y^2 = 1, x = y^2
Other kinds of reflection symmetry you might consider are symmetry with respect to the line y = x, for example, a 1:1 function and its inverse, and symmetry with respect to any given vertical or horizontal line, for example, a parabola is symmetric to a line through its vertex and perpendicular to its directrix.
For more info see
http://faculty.ed.umuc.edu/~swalsh/Math%20Articles/Symmetry.html

The above is a discussion of reflection symmetry in the context of analytic geometry. For more general discussions of reflection symmetry and other kinds of symmetry see
http://en.wikipedia.org/wiki/Reflection_symmetry\
http://mathforum.org/sum95/suzanne/symsusan.html