To determine algebraically whether a function is even, substitute -x into the and simplify. If the resulting is equal to f(x), then the function is even; otherwise, it is not. For example, consider the following function.
f(x) = 3x^4 - 2x^2 + 1
Substitute -x for x and simplify.
f(x) = 3x^4 - 2x^2 + 1
f( -x) = 3( -x)^4 - 2( -x)^2 + 1
f(-x) = 3x^4 - 2(-x)^2 + 1
f(-x) = 3x^4 - 2x^2 + 1
f(-x) = f(x)
Since f(-x)=f(x), the given function is even.