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how to tell if a function is even or odd

How to Tell if a Function Is Even or Odd

Quick scoop: Plug in $$-x$$ wherever you see $$x$$. If nothing changes, the function is even. If every sign flips, it’s odd. If neither happens, it’s neither.

Core idea (the 10‑second test)

For a function f(x)f(x)f(x):
  • It is even if f(−x)=f(x)f(-x)=f(x)f(−x)=f(x) for every xxx.
  • It is odd if f(−x)=−f(x)f(-x)=-f(x)f(−x)=−f(x) for every xxx.
  • If neither condition is true, the function is neither.

Visually:

  • Even: graph is symmetric about the yyy-axis (mirror left ↔ right).
  • Odd: graph has rotational symmetry about the origin (turn 180° around (0,0)(0,0)(0,0)).

Step‑by‑step: algebra test

Use this whenever you’re given a formula for f(x)f(x)f(x).
  1. Start with the function.
    Example: $$f(x) = x^2 - 4$$.
  2. Substitute $$-x$$ for every $$x$$.
    Compute $$f(-x)$$. For the example: $$f(-x) = (-x)^2 - 4 = x^2 - 4$$.
  3. Compare $$f(-x)$$ with $$f(x)$$.
    • If $$f(-x) = f(x)$$ → even.
    • If $$f(-x) = -f(x)$$ → odd.
    • Otherwise → neither.
Mini‑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)=x2+xh(x)=x^2+xh(x)=x2+x
    h(−x)=x2−xh(-x)=x^2-xh(−x)=x2−x. That’s not h(x)h(x)h(x) and not −h(x)-h(x)−h(x) → neither.

Step‑by‑step: graph test

When you only have the graph:
  • Check for even :
    • Fold the graph along the yyy-axis in your mind.
    • If the two sides match exactly, the function is even.
  • Check for odd :
    • Rotate the graph 180∘180^\circ 180∘ around the origin in your mind.
    • If it lands exactly on itself, the function is odd.

Classic shapes:

  • y=x2y=x^2y=x2: U‑shape, symmetric about the yyy-axis → even.
  • y=x3y=x^3y=x3: S‑shape through the origin, symmetric by 180° rotation → odd.
  • y=x3+1y=x^3+1y=x3+1: shifted up; symmetry broken → neither.

Quick reference table (behavior at $$-x$$)

Type Condition Graph symmetry Typical examples
Even $$f(-x) = f(x)$$ Symmetric about $$y$$-axis $$x^2$$, $$x^4$$, $$|x|$$, $$\cos x$$
Odd $$f(-x) = -f(x)$$ Rotational symmetry about origin $$x$$, $$x^3$$, $$\sin x$$, $$\tan x$$
Neither Neither equality holds No simple symmetry $$x^2 + x$$, $$e^x$$

Polynomial shortcut (handy pattern)

For polynomials, look at the powers of xxx:
  • Only even powers (like x2,x4,…x^2,x^4,\dots x2,x4,…) → usually even.
  • Only odd powers , and maybe a constant factor in front → usually odd.
  • A mix of even and odd powers → automatically neither.

Examples:

  • f(x)=3x4−2x2+7f(x)=3x^4-2x^2+7f(x)=3x4−2x2+7 → all even powers, plus constant → even.
  • g(x)=−5x3+2xg(x)=-5x^3+2xg(x)=−5x3+2x → all odd powers → odd.
  • h(x)=x3+x2h(x)=x^3+x^2h(x)=x3+x2 → mix of odd and even → neither.

(You can always confirm by doing the −x-x−x substitution.)

Common questions students have

1. Can a function be both even and odd?

Only the function f(x)=0f(x)=0f(x)=0 (the zero function) is both.
Because 0=−00=-00=−0, it satisfies f(−x)=f(x)f(-x)=f(x)f(−x)=f(x) and f(−x)=−f(x)f(-x)=-f(x)f(−x)=−f(x) at the same time.

2. What about shifting the graph?

  • Vertical shifts (up/down) usually break odd symmetry but can keep even symmetry about the yyy-axis only if they don’t move the center of symmetry.
  • Horizontal shifts (left/right) usually destroy both even and odd symmetry because they move the point or line of symmetry away from the origin and yyy-axis.

One worked example from start to finish

Let’s decide if f(x)=x3−xf(x)=x^3-xf(x)=x3−x is even, odd, or neither.

  1. Compute f(−x)f(-x)f(−x):
    f(−x)=(−x)3−(−x)=−x3+xf(-x)=(-x)^3-(-x)=-x^3+xf(−x)=(−x)3−(−x)=−x3+x.

  2. Factor out −1-1−1:
    −x3+x=−(x3−x)=−f(x)-x^3+x=-(x^3-x)=-f(x)−x3+x=−(x3−x)=−f(x).

  3. Since f(−x)=−f(x)f(-x)=-f(x)f(−x)=−f(x), the function is odd.

Try one yourself:

  • Take g(x)=x4+3g(x)=x^4+3g(x)=x4+3.
  • Compute g(−x)g(-x)g(−x):
    g(−x)=(−x)4+3=x4+3=g(x)g(-x)=(-x)^4+3=x^4+3=g(x)g(−x)=(−x)4+3=x4+3=g(x).
    So ggg is even.

Meta bits for your “Quick Scoop” post

If you’re turning this into a post optimized around “how to tell if a function is even or odd”:
  • Use the substitution rule as your core hook: “Just plug in −x-x−x and compare.”
  • Sprinkle tiny examples: one even (x2x^2x2), one odd (x3x^3x3), one neither (x2+xx^2+xx2+x).
  • Mention graph symmetry: yyy-axis for even, origin rotation for odd.
  • Keep short sections with headings like:
    • “Algebra test in 3 steps”
    • “Graph test in your head”
    • “Polynomial shortcut”
    • “Quick examples”

TL;DR: To tell if a function is even or odd, replace xxx with −x-x−x. If the formula stays identical, it’s even. If every term flips sign, it’s odd. Otherwise, it’s neither.