To determine if a function is even or odd, you check how its output changes when you replace xxx with −x-x−x.

Core idea (Quick Scoop)

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

Geometrically, even functions are symmetric about the y‑axis, and odd functions have rotational symmetry about the origin (a 180∘180^\circ 180∘ turn around (0,0)(0,0)(0,0) gives the same graph).

Step‑by‑step test (algebraic way)

Use this procedure for a function given by a formula, like f(x)=x2−3f(x)=x^2-3f(x)=x2−3 or g(x)=x3−xg(x)=x^3-xg(x)=x3−x.

  1. Start with the function
    • Example: f(x)=x2−3f(x)=x^2-3f(x)=x2−3.
  2. Substitute −x-x−x wherever you see xxx
    • Compute f(−x)f(-x)f(−x).
    • Example: f(−x)=(−x)2−3=x2−3f(-x)=(-x)^2-3=x^2-3f(−x)=(−x)2−3=x2−3.
  1. Compare f(−x)f(-x)f(−x) with f(x)f(x)f(x)
    • 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.
    • Otherwise, it is neither.
  1. State your conclusion
    • For the example above: f(−x)=x2−3=f(x)f(-x)=x^2-3=f(x)f(−x)=x2−3=f(x), so fff is even.

Graphical test (using the graph)

If you only have the graph:

  • A function is even if:
    • The left side is a mirror image of the right side across the y‑axis.
    • Example: The graph of y=x2y=x^2y=x2 is a classic U‑shape symmetric about the y‑axis.
  • A function is odd if:
    • Rotating the graph 180∘180^\circ 180∘ around the origin gives the same graph.
    • Equivalently, every point (x,y)(x,y)(x,y) on the graph has a matching point (−x,−y)(-x,-y)(−x,−y).
    • Example: The graph of y=x3y=x^3y=x3 has this origin symmetry.
  • If the graph has neither type of symmetry, the function is neither even nor odd.

Classic examples (with quick checks)

Polynomial functions

Many textbook functions are standard 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+xh(x)=x^3+xh(x)=x3+x
    • h(−x)=(−x)3+(−x)=−x3−x=−(x3+x)=−h(x)h(-x)=(-x)^3+(-x)=-x^3-x=-(x^3+x)=-h(x)h(−x)=(−x)3+(−x)=−x3−x=−(x3+x)=−h(x) → odd.
  • p(x)=x3+x2p(x)=x^3+x^2p(x)=x3+x2
    • p(−x)=−x3+x2p(-x)=-x^3+x^2p(−x)=−x3+x2, which is neither equal to p(x)p(x)p(x) nor −p(x)-p(x)−p(x) → neither.

A useful pattern:

  • Only even powers of xxx (with no constant shift in the input, like x−1x-1x−1) often give an even function.
  • Only odd powers of xxx and no constant term often give an odd function.
  • Mixing odd and even powers usually gives neither.

Geometric intuition (mini-story style)

Imagine folding the graph along the y‑axis like a sheet of paper:

  • If the left and right halves line up exactly, the function is even.
  • If instead you spin the paper 180∘180^\circ 180∘ around the origin and the graph matches itself, the function is odd.

Any graph that fails both “fold” and “spin” symmetry tests belongs to the messy, but very common, “neither” category.

Quick HTML recap table

Here is a compact HTML table summarizing how to determine if a function is even or odd.

html

<table>
  <thead>
    <tr>
      <th>Type</th>
      <th>Algebraic condition</th>
      <th>Graph symmetry</th>
      <th>Example</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>Even</td>
      <td>f(-x) = f(x) for all x</td>
      <td>Symmetric about the y-axis</td>
      <td>f(x) = x^2, f(x) = cos(x)</td>
    </tr>
    <tr>
      <td>Odd</td>
      <td>f(-x) = -f(x) for all x</td>
      <td>Rotational symmetry about the origin</td>
      <td>f(x) = x^3, f(x) = sin(x)</td>
    </tr>
    <tr>
      <td>Neither</td>
      <td>Neither equation holds for all x</td>
      <td>No y-axis or origin symmetry</td>
      <td>f(x) = x^3 + x^2</td>
    </tr>
  </tbody>
</table>

These criteria are the standard way textbooks and online lessons teach how to determine if a function is even, odd, or neither.

TL;DR:

  • Plug in −x-x−x.
  • 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; otherwise it is neither, and the graph will reflect those symmetries (or lack of them).

Information gathered from public forums or data available on the internet and portrayed here.