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Symmetry was not actually invented. It was just defined after the observation of similarly figured and shaped objects in nature.
Reflection symmetry, reflectional symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection
line symmetry, rotational symmetry, mirror symmetry &liner symmetry
Asymmetry, Radial Symmetry, and Bilateral symmetry.
If a function is even ie if f(-x) = f(x). Such a function would be symmetric about the y-axis. So f(x) is a many-to-one function. The inverse mapping then is one-to-many which is not a function. In fact, the function need not be symmetric about the y-axis. Symmetry about x=k (for any constant k) would also do. Also, leaving aside the question of symmetry, the existence of an inverse depends on the domain over which the original function is defined. Thus, y = f(x) = x2 does not have an inverse if f is defined from the real numbers (R) to R. But if it is defined from (and to) the non-negative Reals there is an inverse - the square-root function.