Parity of Functions

Definition

Geometric Features

• The graph of an even function is symmetric about the y-axis
• The graph of an odd function is symmetric about the origin

Properties

• even × even = even
• odd × odd = even
• even × odd = odd
• even + even = even
• odd + odd = odd
• even + odd is usually neither even nor odd; if one of them is the zero function, the result keeps the parity of the other

Common Examples

• Even functions: $x^2$, $x^4$, $\cos x$, $|x|$
• Odd functions: $x$, $x^3$, $\sin x$, $\tan x$

How to Test Parity

1. Definition method: use the definition directly
2. Graphical method: check whether the graph has the relevant symmetry
3. Algebraic method: compute $f(-x)$ and compare with $f(x)$

Important Properties

• An odd function takes the value 0 at the origin (if the origin is in its domain)
• The derivative of an even function is odd
• The derivative of an odd function is even

---

Exercises

Exercise 1

Determine the parity of $f(x) = x^3 + x$.

Exercise 2

Determine the parity of $f(x) = x^2 + 2x + 1$.

---

Summary

Symbols Used in This Article