Symmetric Functions Examples: 43 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Symmetric 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

A symmetric function is one that remains unchanged (or changes in a predictable way) under specific variable transformations. Even functions satisfy $f(-x) = f(x)$ and are mirror-symmetric about the y-axis; odd functions satisfy $f(-x) = -f(x)$ and have 180-degree rotational symmetry about the origin.

Even functions are symmetric about the y-axis: $f(-x) = f(x)$ . Odd functions have 180° rotational symmetry about the origin: $f(-x) = -f(x)$ .

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: A symmetric function reproduces itself (even) or flips sign (odd) when the input is negated.

Common stuck point: The procedure for symmetric functions is the easy part; the trap is judging symmetry from a rough sketch instead of algebra. Asking "When you replace $x$ with $-x$ , do you get back $f(x)$ (even), $-f(x)$ (odd), or neither?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: When you replace $x$ with $-x$ , do you get back $f(x)$ (even), $-f(x)$ (odd), or neither?

Worked Examples

Example 1

easy

Is the function

f(x, y) = x^2 + y^2 + xy

symmetric in

x

and

y

Answer

Yes, f (x, y) is symmetric in x and y

First step

A function is symmetric in

x

and

y

if swapping

x

and

y

gives the same function:

f(x,y) = f(y,x)

Full solution

2
Compute $f(y,x) = y^2 + x^2 + yx = x^2 + y^2 + xy$ .
3
Since $f(x,y) = f(y,x)$ , the function is symmetric in $x$ and $y$ .

A symmetric function remains unchanged when its variables are interchanged. The elementary symmetric polynomials (like

x+y

xy

x^2+y^2

) are building blocks for all symmetric functions, by the fundamental theorem of symmetric polynomials.

Example 2

medium

Express

x^2 + y^2

in terms of the elementary symmetric polynomials

e_1 = x + y

and

e_2 = xy

Example 3

medium

Express

x^3 + y^3

in terms of

e_1 = x + y

and

e_2 = xy

Example 4

medium

Show that the sum of two even functions is even.

Example 5

medium

f

is odd and continuous at

0

, what must

f(0)

equal?

Example 6

hard

x + y + z = 3

xy + yz + zx = 1

, and

xyz = -1

, find

x^2 + y^2 + z^2

Example 7

hard

Find constants

a, b

so that

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

is odd.

Example 8

challenge

Given roots

x, y

t^2 - 5t + 6 = 0

, find

x^5 + y^5

using Newton's identities.

Practice Problems

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

Example 1

medium

x + y = 7

and

xy = 10

, find

x^3 + y^3

Example 2

hard

Determine whether

f(x, y, z) = x^2y + y^2z + z^2x

is a symmetric function of three variables.

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 + x

even, odd, or neither?

Example 6

easy

f(x) = |x|

even or odd?

Example 7

easy

Is the constant function

f(x) = 7

even or odd?

Example 8

easy

What symmetry does an even function's graph have?

Example 9

easy

What symmetry does an odd function's graph have?

Example 10

easy

f(x) = \cos x

even or odd?

Example 11

medium

Classify

f(x) = x^4 - 3x^2 + 1

Example 12

medium

Classify

f(x) = x^3 - 4x

Example 13

medium

f

is odd, what must

f(0)

equal?

Example 14

medium

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

even, odd, or neither (for

x \ne 0

Example 15

medium

The product of an even function and an odd function is even, odd, or neither?

Example 16

medium

Decompose

f(x) = x^2 + x

into even and odd parts.

Example 17

medium

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

even, odd, or neither?

Example 18

medium

f

is even and

f(3) = 5

, what is

f(-3)

Example 19

medium

Classify

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

Example 20

challenge

Prove that the only function that is both even and odd is

f(x) = 0

Example 21

challenge

f

is odd and

g

is even, classify

h(x) = f(g(x))

Example 22

challenge

Show that

f(x) = x^2 + bx

is even only when

b = 0

Example 23

easy

f(x) = \sin x

even, odd, or neither?

Example 24

easy

Is the product

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

even, odd, or neither?

Example 25

medium

x + y = 5

and

xy = 4

, find

x^2 + y^2

Example 26

medium

x + y = 6

and

x^2 + y^2 = 20

, find

xy

Example 27

medium

Decompose

f(x) = e^x

into its even and odd parts.

Example 28

medium

Is the sum of an even and an odd function (with the odd part not identically zero) ever odd?

Example 29

medium

f

is odd and integrable on

[-a, a]

, find

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

Example 30

medium

f(x, y) = (x-y)^2

a symmetric function of

x

and

y

Example 31

hard

x + y = 4

and

xy = 3

, find

x^4 + y^4

Example 32

hard

Given the polynomial

t^3 - 6t^2 + 11t - 6 = 0

with roots

x, y, z

, find

x^2 + y^2 + z^2

Example 33

hard

f(x)

is odd, prove that

f(x)^2

is even.

Example 34

hard

Determine whether the function

f(x, y, z) = (x-y)(y-z)(z-x)

is symmetric, anti-symmetric, or neither under swaps of variables.

Example 35

challenge

x + y + z = 0

, simplify

x^3 + y^3 + z^3

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Related Concepts

Function Notation Algebraic Symmetry Reflecting Functions

Background Knowledge

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

function notationalgebraic symmetryreflecting functions