Practice Symmetric 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

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).

Showing a random 20 of 50 problems.

Example 1

easy
Is the product f(x)=x2⋅sin⁡x even, odd, or neither?

Example 2

medium
If x+y=5 and xy=4, find x2+y2.

Example 3

medium
Is f(x,y)=(x−y)2 a symmetric function of x and y?

Example 4

medium
Classify f(x)=x3−4x.

Example 5

hard
If x+y+z=3, xy+yz+zx=1, and xyz=−1, find x2+y2+z2.

Example 6

medium
Classify f(x)=x5+x3.

Example 7

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 8

medium
Is f(x)=1x even, odd, or neither (for x≠0)?

Example 9

medium
If f is even and f(3)=5, what is f(−3)?

Example 10

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

Example 11

hard
If f(x) is odd, prove that f(x)2 is even.

Example 12

easy
Is f(x)=cos⁡x even or odd?

Example 13

medium
Decompose f(x)=x2+x into even and odd parts.

Example 14

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

Example 15

easy
If f is even, what symmetry does its graph have about the y-axis?

Example 16

easy
Is the function f(x,y)=x2+y2+xy symmetric in x and y?

Example 17

challenge
Given roots x,y of t2−5t+6=0, find x5+y5 using Newton's identities.

Example 18

easy
Is f(x)=∣x∣ even or odd?

Example 19

easy
Is f(x)=x4−3x2+1 even, odd, or neither?

Example 20

medium
If f is odd and continuous at 0, what must f(0) equal?
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