How to notice symmetry?

Question

What is the easiest method to notice the symmetry of the following function without using any graphical tool: $$g(x)=\frac{1}{\pi \sqrt{4-x^2}}$$

Answer 1

Perhaps the easiest is to note that $$ f(-x) = \frac{1}{\pi \sqrt{4 - (-x)^2}} = \frac{1}{\pi \sqrt{4 - x^2}} = f(x)$$ now if $x > 0$ then by the above we can see that for negative values of $x$ the function maps the same values as the positive side of $x$.

Answer 2

It depends only on $x^2$ so it gives the same value for every positive real $x$ and its negative. Such a function is called even.