Even and Odd Functions Algebraically Flashcards

Even and Odd Functions

• A function is even if f(-x) = f(x) for all x in the domain.
• A function is odd if f(-x) = -f(x) for all x in the domain.
• A function is classified as neither if it does not satisfy the conditions for either even or odd.

Function Classifications

• f(x) = x⁵ + x³ + 1
  • Classification: Neither
• f(x) = x⁵ + x³
  • Classification: Odd
• f(x) = x⁷ - x³ + 5
  • Classification: Neither
• f(x) = x⁶ + x² + 3
  • Classification: Even
• f(x) = x² + 3
  • Classification: Even
• f(x) = (x + 3)³
  • Classification: Neither
• f(x) = |x| + 1
  • Classification: Even
• f(x) = 2x⁷ - 4x
  • Classification: Odd
• f(x) = -x⁵ + x³ + x
  • Classification: Odd
• f(x) = x⁴ + x²
  • Classification: Even
• f(x) = 10x³
  • Classification: Odd
• f(x) = 2x⁵ - 5
  • Classification: Neither

Summary of Classifications

• Functions such as x⁶ + x² + 3, x² + 3, |x| + 1, x⁴ + x², and 10x³ are classified as even functions due to their symmetry about the y-axis.
• Functions like x⁵ + x³, 2x⁷ - 4x, and -x⁵ + x³ + x are classified as odd functions, showing rotational symmetry around the origin.
• Complex functions like x⁵ + x³ + 1 and other forms demonstrated that not all functions fall into even or odd categories; they can be classified as neither.