Even and Odd Functions Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Even and Odd Functions.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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

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

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Even functions give back $f(x)$ when you plug in $-x$ ; odd functions give back $-f(x)$ .

Common stuck point: The procedure for even and odd functions is the easy part; the trap is assuming 'not even' means 'odd'. Asking "Does $f(-x)$ equal $f(x)$ (even), equal $-f(x)$ (odd), or neither?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does $f(-x)$ equal $f(x)$ (even), equal $-f(x)$ (odd), or neither?

Worked Examples

Example 1

easy

Determine whether

f(x) = x^4 - 3x^2 + 2

is even, odd, or neither.

Answer

\text{Even}

First step

Compute

f(-x)

(-x)^4 - 3(-x)^2 + 2 = x^4 - 3x^2 + 2

Full solution

2
Compare: $f(-x) = x^4 - 3x^2 + 2 = f(x)$ .
3
Since $f(-x) = f(x)$ for all $x$ , the function is even.

A function is even if

f(-x) = f(x)

for all

x

in its domain, meaning its graph is symmetric about the

y

-axis. Polynomials with only even powers of

x

(including constant terms, which are

x^0

) are always even functions.

Example 2

medium

Determine whether

g(x) = \frac{x}{x^2 + 1}

is even, odd, or neither.

Example 3

hard

Show that

f(x) = x + \sin(x)

is odd.

Example 4

challenge

Given

f(x) = x^3 + 2x^2 + x + 5

, decompose it into even part

E(x)

and odd part

O(x)

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium

h(x) = x^3 + x^2

even, odd, or neither?

Example 2

hard

Prove that the product of two odd functions is an even function.

Example 3

easy

f(x) = x^2

even, odd, or neither?

Example 4

easy

f(x) = x^3

even, odd, or neither?

Example 5

easy

f(x) = x^2 + 1

even, odd, or neither?

Example 6

easy

f(x) = 2x

even, odd, or neither?

Example 7

easy

f(x) = x^2 + x

even, odd, or neither?

Example 8

easy

f(x) = |x|

even, odd, or neither?

Example 9

easy

f(x) = 5

even, odd, or neither?

Example 10

easy

f(x) = x^4

even, odd, or neither?

Example 11

medium

Classify

f(x) = x^3 - x

Example 12

medium

Classify

f(x) = x^4 - 3x^2 + 2

Example 13

medium

Classify

f(x) = \sin(x)

Example 14

medium

Classify

f(x) = \cos(x)

Example 15

medium

f

is odd and

f(3) = 7

, find

f(-3)

Example 16

medium

f

is even and

f(-2) = 9

, find

f(2)

Example 17

medium

Why is

f(x) = 0

both even and odd?

Example 18

medium

Classify

f(x) = \frac{1}{x}

Example 19

challenge

Classify

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

Example 20

challenge

Classify

f(x) = x\sin(x)

Example 21

medium

f

is even and

g

is odd, classify the product

f\cdot g

Example 22

challenge

Show that any function

f

can be written as a sum of an even and an odd function.

Example 23

easy

f(x) = x^6

even, odd, or neither?

Example 24

easy

f(x) = x^5

even, odd, or neither?

Example 25

easy

f(x) = 4x^3 - x

even, odd, or neither?

Example 26

easy

f(x) = x^2 + 2x + 1

even, odd, or neither?

Example 27

easy

Classify

f(x) = \tan(x)

Example 28

easy

Classify

f(x) = \sec(x)

Example 29

medium

Classify

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

Example 30

medium

Classify

f(x) = \frac{x}{x^4 + 2}

Example 31

medium

f

is even and

g

is even, classify

f + g

Example 32

medium

f

is odd and

g

is odd, classify

f + g

Example 33

medium

f

is even and

g

is even, classify the product

f \cdot g

Example 34

medium

Classify

f(x) = e^x + e^{-x}

Example 35

medium

Classify

f(x) = e^x - e^{-x}

Example 36

hard

Classify

f(x) = x^2 \sin(x)

Example 37

hard

Classify

f(x) = \ln\!\left(\frac{1+x}{1-x}\right)

(-1, 1)

Example 38

hard

f

is even and

f(2) = 3

, what is

f(2) + f(-2)

Example 39

hard

f

is odd and continuous on

[-a, a]

, what is

\int_{-a}^{a} f(x)\,dx

Example 40

hard

Suppose

f(x) = ax^4 + bx^3 + cx^2 + dx + e

is even. What must

b

and

d

equal?

Example 41

challenge

Let

f(x) = e^x

. Find its even part

\frac{f(x)+f(-x)}{2}

Example 42

challenge

f

is even and

g

is odd, classify the composition

f \circ g

(i.e.,

f(g(x))

← Back to Even and Odd Functions Practice Mode

Related Concepts

Function Notation Reflecting Functions Algebraic Symmetry

Background Knowledge

These ideas may be useful before you work through the harder examples.

function notationreflecting functionsalgebraic symmetry