Practice Even and Odd Functions in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

An even function satisfies f(−x)=f(x)f(-x) = f(x) (symmetric about yy-axis); an odd function satisfies f(−x)=−f(x)f(-x) = -f(x) (rotational symmetry about origin).

Even means mirror across yy-axis; odd means rotational symmetry through the origin.

Showing a random 20 of 50 problems.

Example 1

medium
Classify f(x)=x3−xf(x) = x^3 - x.

Example 2

easy
Is f(x)=x2+2x+1f(x) = x^2 + 2x + 1 even, odd, or neither?

Example 3

medium
Classify f(x)=cos⁡(x)f(x) = \cos(x).

Example 4

medium
Classify f(x)=sin⁡(x)f(x) = \sin(x).

Example 5

challenge
Given f(x)=x3+2x2+x+5f(x) = x^3 + 2x^2 + x + 5, decompose it into even part E(x)E(x) and odd part O(x)O(x).

Example 6

challenge
Show that any function ff can be written as a sum of an even and an odd function.

Example 7

hard
Classify f(x)=x2sin⁡(x)f(x) = x^2 \sin(x).

Example 8

easy
Is f(x)=x2f(x) = x^2 even, odd, or neither?

Example 9

challenge
If ff is even and gg is odd, classify the composition f∘gf \circ g (i.e., f(g(x))f(g(x))).

Example 10

easy
Is f(x)=−xf(x) = -x even, odd, or neither?

Example 11

easy
Is f(x)=x6f(x) = x^6 even, odd, or neither?

Example 12

medium
If ff is odd and f(3)=7f(3) = 7, find f(−3)f(-3).

Example 13

medium
If ff is even and gg is even, classify the product f⋅gf \cdot g.

Example 14

easy
Classify f(x)=tan⁡(x)f(x) = \tan(x).

Example 15

medium
If ff is even and f(−2)=9f(-2) = 9, find f(2)f(2).

Example 16

easy
Is f(x)=5f(x) = 5 even, odd, or neither?

Example 17

medium
If ff is even, what does its graph have? (Pick one: yy-axis symmetry / origin symmetry.)

Example 18

easy
Is f(x)=4x3−xf(x) = 4x^3 - x even, odd, or neither?

Example 19

easy
Is f(x)=∣x∣f(x) = |x| even, odd, or neither?

Example 20

easy
Is f(x)=x5f(x) = x^5 even, odd, or neither?
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