How to determine whether the given function is even, odd, or neither?

f (x) = x^3 + x^2 + 3

f (x) = x^3 + x^2 + 3x^0 is neither even nor odd. Why? Because we have a mixture of even and odd powers of x here: x^3 (odd) and x^2 and x^0 (even).

Even functions: all component functions are even (e. g., x^2 and x^0).

Odd functions: all component functions are odd (e. g., x^5 and x^3.

Neither: There's a mixture of even ad odd functions.

Another way to test for even, odd or neither:

Even functions: f (-x) = f (x). Changing the sign of the input (x) doesn't change the sign of the output.

Odd functions: f (-x) = - f (x). Changing the sign of the input changes the sign of the output.