Function Families Math Example 4

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Example 4

hard
The family y=xny=x^n for integer n≥1n\geq1 has different behavior depending on whether nn is even or odd. Classify and explain the symmetry of y=x2y=x^2, y=x3y=x^3, y=x4y=x^4, y=x5y=x^5.

Solution

  1. 1
    Even nn: y=x2y=x^2 and y=x4y=x^4 satisfy f(−x)=f(x)f(-x)=f(x) — even functions, symmetric about yy-axis. Both open upward.
  2. 2
    Odd nn: y=x3y=x^3 and y=x5y=x^5 satisfy f(−x)=−f(x)f(-x)=-f(x) — odd functions, symmetric about origin. Both pass from third to first quadrant.
  3. 3
    Pattern: power functions with even exponent are even; with odd exponent, odd. As nn increases, the curve is flatter near 00 and steeper for ∣x∣>1|x|>1.

Answer

Even nn: even functions (y-axis symmetry); Odd nn: odd functions (origin symmetry)
The parity of the exponent determines the symmetry of the power function. This is a fundamental organizing principle of the power function family, and it extends to all monomials axnax^n.

About Function Families

A function family is a group of functions sharing the same general form and behavior, differing only in the values of one or more parameters.

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