Symmetric Functions Examples in Math

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

Symmetric functions are unchanged under specific variable swaps or sign transformations.

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: Even/odd symmetry halves the work — knowing f on [0, \infty) completely determines f on all of \mathbb{R} for both even and odd functions.

Common stuck point: Students test symmetry visually without checking algebraic conditions.

Sense of Study hint: Substitute -x (or swap variables) and compare with the original expression.

Worked Examples

Example 1

easy
Is the function f(x, y) = x^2 + y^2 + xy symmetric in x and y?

Solution

  1. 1
    A function is symmetric in x and y if swapping x and y gives the same function: f(x,y) = f(y,x).
  2. 2
    Compute f(y,x) = y^2 + x^2 + yx = x^2 + y^2 + xy.
  3. 3
    Since f(x,y) = f(y,x), the function is symmetric in x and y.

Answer

\text{Yes, } f(x,y) \text{ is symmetric in } x \text{ 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.

Practice Problems

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

Example 1

medium
If 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.
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Background Knowledge

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

function notationalgebraic symmetryreflecting functions