Even, Odd or Neither? Test a Function by Replacing x with −x

On a domain symmetric about zero, an even function satisfies f(−x) = f(x), while an odd function satisfies f(−x) = −f(x). If neither identity holds, the function is neither.

Check the domain before the formula

The test needs x and −x to be allowed together. For example, √x with its usual real domain [0, ∞) is not even or odd on that domain, because a positive input’s negative is not permitted. A graph drawn only on one side cannot establish the required symmetry.

An even function has y-axis symmetry. An odd function has symmetry under a half-turn about the origin. These are properties of the complete function, including its domain, not merely of the largest exponent in an expression.

Worked example: an odd polynomial

Take f(x) = x³ − 4x on all real numbers. Substitute −x into every occurrence of x: f(−x) = (−x)³ − 4(−x) = −x³ + 4x. Factor out −1 to get −(x³ − 4x) = −f(x). Therefore f is odd.

For g(x) = x⁴ + 2x², both powers remain unchanged under x → −x. Hence g(−x) = g(x), so g is even. For h(x) = x² + x, substitution gives x² − x, which equals neither h(x) nor −h(x) as an identity. Thus h is neither.

One numerical pair can disprove, not prove

If h(2) = 6 and h(−2) = 2, the pair disproves evenness because the values differ, and disproves oddness because 2 ≠ −6. However, a few matching pairs cannot prove a general identity. Use algebra to establish the property for every permitted input.

The zero function on a symmetric domain is both even and odd: zero equals its own negative. For other functions, an odd function defined at zero must have f(0) = 0. That is a necessary condition, not a sufficient one. A function can pass through the origin and still be neither.

Common mistake

Do not assume every expression containing an even power is even. A linear term or constant can change the classification. Rational functions also require domain checks: 1/x is odd on the nonzero real numbers, a symmetric domain even though zero is excluded.

When writing your conclusion, include the algebraic comparison and the domain. The phrase “all powers are odd” is only a convenient observation for a polynomial with no constant term, not a universal definition.

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Three practice questions with worked answers

1. Classify f(x) = x⁶ + 3 over the reals.

   f(−x) = x⁶ + 3 = f(x), and the domain is symmetric. The function is even.

2. Classify g(x) = 2x⁵ − x over the reals.

   g(−x) = −2x⁵ + x = −g(x). The function is odd, with g(0) = 0 as a consistency check.

3. Classify h(x) = x³ + 1 over the reals.

   h(−x) = −x³ + 1. It is neither x³ + 1 nor −x³ − 1 as an identity, so the function is neither even nor odd.

Related study guides

• Domain and Range from a Function Rule
• Sets, Relations and Functions Class 11 Notes with Examples

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References and authorship

Original examples and practice questions, prepared with AI assistance and editorial arithmetic review. Related definitions and methods: OpenStax: Review of Functions. No textbook or exam-board affiliation is claimed.