A function is called even if it satisfies the condition f(−x)=f(x)f(-x)=f(x)f(−x)=f(x) for every xxx in its domain, meaning the function’s values are symmetric about the y-axis. A function is called odd if it satisfies f(−x)=−f(x)f(-x)=-f(x)f(−x)=−f(x) for every xxx in its domain, which implies symmetry about the origin.
Even Functions
- The output does not change when the input sign is reversed.
- Graphically symmetric about the y-axis.
- Example: f(x)=x2f(x)=x^2f(x)=x2, since f(−x)=(−x)2=x2=f(x)f(-x)=(-x)^2=x^2=f(x)f(−x)=(−x)2=x2=f(x).
Odd Functions
- The output changes sign when the input sign is reversed.
- Graphically symmetric about the origin.
- Example: f(x)=x3f(x)=x^3f(x)=x3, since f(−x)=(−x)3=−x3=−f(x)f(-x)=(-x)^3=-x^3=-f(x)f(−x)=(−x)3=−x3=−f(x).
How to Determine
- Substitute −x-x−x into the function.
- If f(−x)=f(x)f(-x)=f(x)f(−x)=f(x), the function is even.
- If f(−x)=−f(x)f(-x)=-f(x)f(−x)=−f(x), the function is odd.
- If neither, the function is neither even nor odd.
The symmetry properties help visually identify the function type, with even functions reflecting across the y-axis and odd functions having rotational symmetry around the origin.