Even and Odd Functions

Even and Odd Functions

• Work out $f\left(-x\right)$ first, and compare it to $f(x)$. If they are the same, the function is even.
• If not, find $-f\left(x\right)$ next and compare it to $f\left(-x\right)$. If they are the same, the function is odd. If not, the function is neither even nor odd.

Properties

• A function can be even, odd, or neither.
• The sum of two even functions is even. The sum of two odd functions is odd.
• The sum of an even and an odd function is neither even nor odd (unless one of them is zero).
• The product of two even or two odd functions is even.
• The product of an even and an odd function is odd.
• The reciprocal of an even/odd function is even/odd.

• The function $y=x^2$ is even because for any x, we have: $f\left(x\right)=x^2=\left(-x\right)^2=f\left(-x\right)$
• The function $y=x^3$ is odd because for any x, we have: $f\left(-x\right)=\left(-x\right)^3=-x^3=-f\left(x\right)$