Function Symmetry: Even & Odd Functions

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Determine whether a function is even, odd, or neither.
  • Use algebraic tests involving \(f(-x)\).
  • Recognize symmetry visually from graphs.
  • Understand how symmetry shows up in standardized test questions.

Key Ideas

Some functions have special symmetry that makes them easier to analyze.

Even Functions

A function is even if: \[ f(-x) = f(x) \]

Graphically: the graph is symmetric across the \(y\)-axis.

Common examples:

  • \(f(x) = x^2\)
  • \(f(x) = |x|\)
  • \(f(x) = \cos x\)

Graph of \(y = x^2\) showing symmetry across the \(y\)-axis.

Odd Functions

A function is odd if: \[ f(-x) = -f(x) \]

Graphically: the graph is symmetric about the origin
(rotate the graph 180° about the origin).

Common examples:

  • \(f(x) = x^3\)
  • \(f(x) = x\)
  • \(f(x) = \sin x\)

Graph of \(y = x^3\) showing symmetry about the origin.

Neither

Many functions are neither even nor odd.
For example: \(x^2 + x\), \(e^x\), or any graph without symmetry.

Common Problem Types

1. Algebraic Test

Compute \(f(-x)\) and compare it with \(f(x)\) or \(-f(x)\).

2. Graph-Based Identification

Check for \(y\)-axis symmetry (even) or origin symmetry (odd).

3. Table or Mapping Tests

Look for matching or opposite pairs.

4. Domain Considerations

A function cannot be even or odd unless its domain is symmetric about 0.

Strategies

  • Always use parentheses when computing \(f(-x)\).
  • Check the domain—square roots and logs often fail symmetry due to domain limits.
  • For graphs:
    • Even → mirror across \(y\)-axis
    • Odd → rotate 180° around the origin
  • If neither comparison works, the function is neither.

Worked Examples

Example 1 — Algebraic Test (Even)

Check if: \[ f(x) = x^2 - 4 \]

Compute: \[ \begin{split} f(-x) &= (-x)^2 - 4 \\ &= x^2 - 4 \end{split} \]

Since \(f(-x) = f(x)\), the function is even.

Example 2 — Algebraic Test (Odd)

Check if: \[ g(x) = x^3 + x \]

Compute: \[ \begin{split} g(-x) &= (-x)^3 + (-x) \\ &= -x^3 - x \\ &= -(x^3 + x) \end{split} \]

Since \(g(-x) = -g(x)\), the function is odd.

Example 3 — Graph Test

If a graph mirrors perfectly across the \(y\)-axis, the function is even.

Graph demonstrating left–right mirror symmetry across the \(y\)-axis.
WarningCommon Mistakes
  • Thinking “even = even powers” and “odd = odd powers.”
    (Not always true—mixed terms break symmetry.)
  • Forgetting parentheses when evaluating \(f(-x)\).
  • Assuming symmetry without checking.
  • Ignoring domain: a function like \(\sqrt{x}\) cannot be even or odd.

Practice Problems

  1. Is \(f(x) = x^4 - 7\) even, odd, or neither?
  2. Test \(g(x) = x^3 - 2x\).
  3. Is \(h(x) = \sqrt{x}\) even, odd, or neither?
  4. A graph is symmetric about the origin. What type of function is it?
  5. If \(f\) is even, which must be true?
    • A. \(f(-x) = -f(x)\)
    • B. \(f(-x) = f(x)\)
    • C. \(f(x) = x\)

1.
Compute \(f(-x) = (-x)^4 - 7 = x^4 - 7\) → even

2.
\[ g(-x) = (-x)^3 - 2(-x) = -x^3 + 2x = -(x^3 - 2x) \]
→ odd

3.
Domain of \(\sqrt{x}\) is \(x \ge 0\) → not symmetric → neither

4.
Origin symmetry → odd function

5.
Correct statement: B

Summary

  • Even functions satisfy \(f(-x) = f(x)\) and show \(y\)-axis symmetry.
  • Odd functions satisfy \(f(-x) = -f(x)\) and show origin symmetry.
  • Many functions are neither—mixed parity or restricted domains break symmetry.
  • Use algebraic tests for certainty; visual checks support reasoning.
  • Use parentheses every time you evaluate \(f(-x)\).
  • Check graph symmetry: mirror (even) vs. rotation (odd).
  • Domain must include both \(x\) and \(-x\) to qualify.
  • Mixed terms often make a function neither even nor odd.