Practice Symmetric Functions in Math

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

Quick 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).

Example 1

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

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
If x + y = 7 and xy = 10, find x^3 + y^3.

Example 4

hard
Determine whether f(x, y, z) = x^2y + y^2z + z^2x is a symmetric function of three variables.
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