how to know if a function is even or odd

To know if a function is even or odd, you use the following criteria based on substituting -x into the function:

• A function f(x)f(x)f(x) is even if f(−x)=f(x)f(-x)=f(x)f(−x)=f(x) for all xxx in the domain. Even functions have symmetry about the y-axis.
• A function f(x)f(x)f(x) is odd if f(−x)=−f(x)f(-x)=-f(x)f(−x)=−f(x) for all xxx in the domain. Odd functions have symmetry about the origin.
• If neither of these conditions are met, the function is neither even nor odd.

Graphically, even functions look symmetric when reflected across the y-axis, and odd functions look symmetric when rotated 180 degrees about the origin. Algebraically, you check by substituting −x-x−x into the function and comparing:

• If you get the original function back, it is even.
• If you get the negative of the function, it is odd.
• Otherwise, it is neither.

This approach works for any type of function and is the formal way to determine evenness or oddness.