There are many reasons why we call those even and odd functions. One is that the taylor series of an even function only includes even powers whereas the Taylor series of an odd function only includes odd power.
But to recognize when and what type of symmetry there is, the most useful property is that they are called even/odd because they often act similar to even/odd numbers:
- The sum of two even functions is even, and any constant multiple of an even function is even.
- The sum of two odd functions is odd, and any constant multiple of an odd function is odd.
- The difference between two odd functions is odd.
- The difference between two even functions is even.
- The product of two even functions is an even function.
- The product of two odd functions is an even function.
- The product of an even function and an odd function is an odd function.
- The quotient of two even functions is an even function.
- The quotient of two odd functions is an even function.
- The quotient of an even function and an odd function is an odd function.
With that in mind, you need to have a couple basic even/odd functions in your sleeve.
$x^n$ is even for $n$ even and odd for $n$ odd (even for negative integers). the function $\cos$ is even while $\sin$ is odd. What would $\tan$ be? The absolute value $|x|$ is even.
When looking at some functions $f(x)=g(x)/h(x)$, use the above properties to determine the "parity" of $g(x)$ and $h(x)$ and then check their quotient.
Note however that some functions are neither even nor odd, for example $x^3-x+1$.