Symmetry (Meta) Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy
Determine whether f(x)=x3f(x) = x^3 is odd, even, or neither, by testing the symmetry condition.

Solution

  1. 1
    Even function test: f(−x)=(−x)3=−x3=−f(x)≠f(x)f(-x) = (-x)^3 = -x^3 = -f(x) \ne f(x) (unless x=0x=0). Not even.
  2. 2
    Odd function test: f(−x)=−x3=−f(x)f(-x) = -x^3 = -f(x). This holds for all xx. Odd function.

Answer

f(x)=x3 is an odd function (symmetric about the origin)f(x)=x^3 \text{ is an odd function (symmetric about the origin)}
An odd function satisfies f(−x)=−f(x)f(-x)=-f(x), meaning its graph has rotational symmetry of 180°180° about the origin. An even function satisfies f(−x)=f(x)f(-x)=f(x), meaning it is symmetric about the yy-axis.

About Symmetry (Meta)

A property of a mathematical object that remains unchanged under a specified transformation — reflection, rotation, translation, or algebraic substitution.

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More Symmetry (Meta) Examples

Example 1 easy

Show that the equation [formula] is symmetric about both coordinate axes and the origin. Verify by s

Example 2 medium

Use symmetry to evaluate [formula] for even [formula].

Example 4 medium

Use the symmetry of [formula] (odd function) and [formula] (even function) to simplify: [formula].

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