Even and Odd Functions

Fold a sheet of paper in half, drop some ink on it and unfold it again. You get a picture whose left half is the mirror image of the right half. Some graphs of functions look exactly like that — they are symmetric about the y-axis. We call such functions even. Other graphs land exactly on themselves when you turn them half a turn around the point (0, 0). Those are odd functions.

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Contents

• Even functions
• Odd functions
• How to check by calculation
• Neither even nor odd
• Watch the domain
• Common mistakes
• Summary
• Try it yourself
• Practice

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Even functions

Take f(x) = x². Substitute 3 and −3:

• f(3) = 3² = 9,
• f(−3) = (−3)² = 9.

Opposite numbers gave the same value. This works for every x, because squaring "eats" the sign.

A function f is even if for every x in its domain:

f(−x) = f(x) — read "f of minus x equals f of x".

The graph of an even function is symmetric about the y-axis. A point (a, b) on the graph always has a "twin" (−a, b).

Examples of even functions: x², x⁴, |x|, x² − 5, 3/(x² + 1).

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Odd functions

Take f(x) = x³:

• f(2) = 2³ = 8,
• f(−2) = (−2)³ = −8.

Opposite numbers gave opposite values.

A function f is odd if for every x in its domain:

f(−x) = −f(x) — read "f of minus x equals minus f of x".

The graph of an odd function is symmetric about the origin (0, 0). A point (a, b) has a "twin" (−a, −b).

Examples of odd functions: x, x³, x⁵, 1/x, x³ − 4x.

💡 The names are not random: x² and x⁴ have an even exponent and are even functions; x and x³ have an odd exponent and are odd functions.

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How to check by calculation

The method has three steps:

1. Check that the domain is symmetric about zero (whenever it contains x, it also contains −x).
2. Substitute −x for x in the formula and simplify.
3. Compare the result with f(x) and with −f(x).

Worked example 1

f(x) = x⁴ − 3x².

1. The domain is ℝ, which is symmetric.
2. f(−x) = (−x)⁴ − 3(−x)² = x⁴ − 3x².
3. We got the original formula: f(−x) = f(x). The function is even.

Worked example 2

f(x) = x³ − 5x.

1. The domain is ℝ.
2. f(−x) = (−x)³ − 5(−x) = −x³ + 5x.
3. That is −(x³ − 5x) = −f(x). The function is odd.

Worked example 3

f(x) = $\frac{2x}{x^2+1}$.

1. The denominator x² + 1 is never zero, so the domain is ℝ.
2. f(−x) = $\frac{2(-x)}{(-x{)}^2+1}=\frac{-2x}{x^2+1}$.
3. That is −f(x). The function is odd.

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Neither even nor odd

Most functions are neither even nor odd.

Worked example 4

f(x) = x² + x.

• f(2) = 4 + 2 = 6,
• f(−2) = 4 − 2 = 2.

The value 2 is not equal to 6 (so it is not even) and not equal to −6 (so it is not odd). One such example is enough to disprove it.

💡 Typical sign: the formula mixes even and odd powers (x² + x), or the graph is shifted sideways, for example |x − 2| or (x − 1)².

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Watch the domain

The function f(x) = x² for x ∈ [−2, 3] is not even. The number 3 is in the domain, but −3 is not. There is nothing to compare with, so the equation f(−3) = f(3) cannot hold.

If the domain is not symmetric about zero, the function is neither even nor odd — whatever the formula is.

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Common mistakes

• (−x)² = −x². No! (−x)² = x², the minus disappears when you square. But (−x)³ = −x³.
• Checking only one number. Being even must hold for all x. One example is only enough to show that a function is not even (or not odd).
• Forgetting the constant. x³ + 2 is not odd: f(−x) = −x³ + 2, but −f(x) = −x³ − 2.
• Ignoring the domain. A domain that is not symmetric means neither even nor odd.

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Summary

• Even: f(−x) = f(x), graph symmetric about the y-axis.
• Odd: f(−x) = −f(x), graph symmetric about the point (0, 0).
• First check that the domain is symmetric about zero.
• Substitute −x, simplify, and compare with f(x) and −f(x).

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Try it yourself

Decide whether each function is even, odd or neither:

1. f(x) = 2x⁶ − x²
2. f(x) = x⁵ + x
3. f(x) = x³ + 1
4. f(x) = |x| − 4
5. f(x) = x² for x ∈ [−3, 3]

Answers: 1. even, 2. odd, 3. neither, 4. even, 5. even (the domain is symmetric).

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Practice

• 🔍 Even and odd functions
• 📊 Function properties from a graph

👉 Next article: Quadratic Functions and Their Graphs