To determine if a function is even or odd, you use the following algebraic tests:

• A function f(x)f(x)f(x) is even if for every xxx in its domain, f(−x)=f(x)f(-x)=f(x)f(−x)=f(x). This means the function's graph is symmetric about the y-axis.
• A function f(x)f(x)f(x) is odd if for every xxx in its domain, f(−x)=−f(x)f(-x)=-f(x)f(−x)=−f(x). This means the graph is symmetric about the origin.
• If f(−x)f(-x)f(−x) is neither equal to f(x)f(x)f(x) nor −f(x)-f(x)−f(x), the function is neither even nor odd.

The practical step is to substitute −x-x−x into the function in place of xxx. If the resulting expression simplifies back to the original function f(x)f(x)f(x), the function is even. If it simplifies to the negative of the original function, it's odd. Otherwise, it's neither. Graphically, even functions exhibit mirror symmetry about the y-axis, and odd functions have rotational symmetry about the origin.