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Symmetry of Functions

Symmetry of Functions. Even, Odd, or Neither?. Even Functions. All exponents are even. May contain a constant. f(x) = f(-x) Symmetric about the y-axis. All even exponents. Example:. Both exponents are even. It does not matter what the coefficients are. May Contain a Constant. Example.

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Symmetry of Functions

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  1. Symmetry of Functions Even, Odd, or Neither?

  2. Even Functions • All exponents are even. • May contain a constant. • f(x) = f(-x) • Symmetric about the y-axis

  3. All even exponents • Example: Both exponents are even. It does not matter what the coefficients are.

  4. May Contain a Constant • Example Even exponents (coefficients don’t matter) Constant does not affect even function.

  5. f(x) = f(-x) • Given f(x) = 5x² - 7, find f(-x) to determine if f(x) is even, odd, or neither. • Substitute –x for x. • f(-x) = 5(-x)² - 7 = 5x² -7 • Because f(x) = f(-x), f(x) = 5x² - 7 is an even function.

  6. f(x) = f(-x) • Given f(x) = 4x² - 2x + 1, determine if f(x) is even, odd, or neither. • Substitute –x for x. • f(-x) = 4(-x)² - 2(-x) + 1 = 4x² +2x + 1 • f(-x) ≠ f(x) • Therefore, f(x) is NOT an even function. (We will revisit to determine what it is.)

  7. Symmetric About the y-axis • The following are symmetric about the y-axis.

  8. Odd Functions • Only odd exponents. • NO constants! • f(-x) = -f(x) • Symmetric about the origin.

  9. All Odd Exponents • Example Understood 1 exponent All odd exponents.

  10. NO Constants • Example: Odd exponents NO constants in odd functions!

  11. f(-x) = -f(x) • Given f(x) = 4x³ + 2x, find f(-x) and f(-x) to determine if f(x) is even, odd, or neither. • f(-x) = 4(-x)³ + 2(-x) = -4x³ - 2x • -f(x) = -4x³ - 2x • Because f(-x) = -f(x), f(x) is an odd function.

  12. f(-x) = -f(x) • Given f(x) = 5x³ + 7x², find f(-x) and f(-x) to determine if f(x) is even, odd, or neither. • f(-x) = 5(-x)³ + 7(-x)² = -5x³ + 7x² • -f(x) = -5x³ - 7x² • f(-x) ≠ -f(x), therefore f(x) is NOT an odd function.

  13. Symmetric About the Origin • These graphs are symmetric about the origin.

  14. Neither? • Mixture of even and odd exponents. • All odd exponents with a constant. • f(x) ≠ f(-x) AND f(-x) ≠ -f(x)

  15. Examples of Neither • f(x) = 4x³ - 5x² • f(x) = 5x³ + 7 Mixture of odd and even exponents. Odd exponents with a constant.

  16. Examples of Neither • If f(x) = -3x³ + 2x², determine if f(x) is even, odd, or neither. • Find f(-x). • f(-x) = -3(-x)³ + 2(-x)² = 3x³ + 2x² • Find –f(x). • -f(x) = 3x³ - 2x² • Because f(-x) ≠ f(x) and f(-x) ≠ -f(x), f(x) is neither even nor odd.

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