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what makes a function even or odd

A function is even if its values mirror perfectly left and right of the y-axis, and odd if its values mirror with a sign flip around the origin.

Core idea

  • Even: f(−x)=f(x)f(-x)=f(x)f(−x)=f(x) for every xxx in the domain.
  • Odd: f(−x)=−f(x)f(-x)=-f(x)f(−x)=−f(x) for every xxx in the domain.
  • If neither equation holds for all xxx, the function is neither even nor odd.

Visual picture

  • Even function graphs are symmetric across the y-axis: folding the graph along the y-axis makes the two sides overlap.
* Example: f(x)=x2f(x)=x^2f(x)=x2 or f(x)=cos⁡(x)f(x)=\cos(x)f(x)=cos(x).
  • Odd function graphs are symmetric about the origin: rotate the graph 180∘180^\circ 180∘ around the origin and it overlaps itself.
* Example: f(x)=x3f(x)=x^3f(x)=x3 or f(x)=sin⁡(x)f(x)=\sin(x)f(x)=sin(x).

Quick test procedure

  1. Start with your function f(x)f(x)f(x).
  2. Compute f(−x)f(-x)f(−x) by plugging −x-x−x in place of xxx.
  1. Compare:
    • If the result simplifies to exactly f(x)f(x)f(x), the function is even.
 * If the result simplifies to −f(x)-f(x)−f(x), the function is **odd**.

 * Otherwise, it is **neither**.

Simple examples

  • f(x)=x2f(x)=x^2f(x)=x2:
    • f(−x)=(−x)2=x2=f(x)f(-x)=(-x)^2=x^2=f(x)f(−x)=(−x)2=x2=f(x) → even.
  • g(x)=x3g(x)=x^3g(x)=x3:
    • g(−x)=(−x)3=−x3=−g(x)g(-x)=(-x)^3=-x^3=-g(x)g(−x)=(−x)3=−x3=−g(x) → odd.
  • h(x)=x3+1h(x)=x^3+1h(x)=x3+1:
    • h(−x)=−x3+1h(-x)=-x^3+1h(−x)=−x3+1 (not equal to h(x)h(x)h(x) or −h(x)-h(x)−h(x)) → neither.

Extra neat fact

Any function on a symmetric domain (like (−∞,∞)(-\infty,\infty)(−∞,∞)) can be split into an even part and an odd part, which is why these types are so useful in higher math and physics.

TL;DR: “Even” means f(−x)f(-x)f(−x) matches f(x)f(x)f(x) and the graph reflects across the y-axis, while “odd” means f(−x)f(-x)f(−x) is the negative of f(x)f(x)f(x) and the graph has origin symmetry.