symmetry of functions even, odd, or neither?. even functions all exponents are even. may contain a...
TRANSCRIPT
![Page 1: 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](https://reader036.vdocument.in/reader036/viewer/2022081002/56649dce5503460f94ac29aa/html5/thumbnails/1.jpg)
Symmetry of Functions
Even, Odd, or Neither?
![Page 2: 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](https://reader036.vdocument.in/reader036/viewer/2022081002/56649dce5503460f94ac29aa/html5/thumbnails/2.jpg)
Even Functions
• All exponents are even.
• May contain a constant.
• f(x) = f(-x)
• Symmetric about the y-axis
![Page 3: 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](https://reader036.vdocument.in/reader036/viewer/2022081002/56649dce5503460f94ac29aa/html5/thumbnails/3.jpg)
All even exponents
• Example:
4 2( ) 5 3f x x x
Both exponents are even. It does not matter what the coefficients are.
![Page 4: 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](https://reader036.vdocument.in/reader036/viewer/2022081002/56649dce5503460f94ac29aa/html5/thumbnails/4.jpg)
May Contain a Constant
• Example
6 2( ) 3 7 1f x x x
Even exponents (coefficients don’t matter)
Constant does not affect even function.
![Page 5: 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](https://reader036.vdocument.in/reader036/viewer/2022081002/56649dce5503460f94ac29aa/html5/thumbnails/5.jpg)
f(x) = f(-x)
• Given f(x) = 5x² - 7, find f(-x) to determine if f(x) is even, odd, or neither.
1) Substitute –x for x.
2) f(-x) = 5(-x)² - 7 = 5x² -7
3) Because f(x) = f(-x),
f(x) = 5x² - 7 is an even function.
![Page 6: 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](https://reader036.vdocument.in/reader036/viewer/2022081002/56649dce5503460f94ac29aa/html5/thumbnails/6.jpg)
f(x) = f(-x)
• Given f(x) = 4x² - 2x + 1, determine if f(x) is even, odd, or neither.
1) Substitute –x for x.
2) f(-x) = 4(-x)² - 2(-x) + 1 = 4x² +2x + 1
3) f(-x) ≠ f(x)
4) Therefore, f(x) is NOT an even function. (We will revisit to determine what it is.)
![Page 7: 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](https://reader036.vdocument.in/reader036/viewer/2022081002/56649dce5503460f94ac29aa/html5/thumbnails/7.jpg)
Symmetric About the y-axis
• The following are symmetric about the y-axis.
![Page 8: 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](https://reader036.vdocument.in/reader036/viewer/2022081002/56649dce5503460f94ac29aa/html5/thumbnails/8.jpg)
Odd Functions
• Only odd exponents.
• NO constants!
• f(-x) = -f(x)
• Symmetric about the origin.
![Page 9: 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](https://reader036.vdocument.in/reader036/viewer/2022081002/56649dce5503460f94ac29aa/html5/thumbnails/9.jpg)
All Odd Exponents
• Example
5 3( ) 8 6 4f x x x x
All odd exponents.
Understood 1 exponent
![Page 10: 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](https://reader036.vdocument.in/reader036/viewer/2022081002/56649dce5503460f94ac29aa/html5/thumbnails/10.jpg)
NO Constants• Example:
5 3( ) 6 8 5f x x x Odd exponents
NO constants in odd functions!
![Page 11: 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](https://reader036.vdocument.in/reader036/viewer/2022081002/56649dce5503460f94ac29aa/html5/thumbnails/11.jpg)
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.
![Page 12: 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](https://reader036.vdocument.in/reader036/viewer/2022081002/56649dce5503460f94ac29aa/html5/thumbnails/12.jpg)
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.
![Page 13: 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](https://reader036.vdocument.in/reader036/viewer/2022081002/56649dce5503460f94ac29aa/html5/thumbnails/13.jpg)
Symmetric About the Origin
• These graphs are symmetric about the origin.
![Page 14: 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](https://reader036.vdocument.in/reader036/viewer/2022081002/56649dce5503460f94ac29aa/html5/thumbnails/14.jpg)
Neither?
• Mixture of even and odd exponents.
• All odd exponents with a constant.
• f(x) ≠ f(-x) AND f(-x) ≠ -f(x)
![Page 15: 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](https://reader036.vdocument.in/reader036/viewer/2022081002/56649dce5503460f94ac29aa/html5/thumbnails/15.jpg)
Examples of Neither• f(x) = 4x³ - 5x²
• f(x) = 5x³ + 7
Mixture of odd and even exponents.
Odd exponents with a constant.
![Page 16: 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](https://reader036.vdocument.in/reader036/viewer/2022081002/56649dce5503460f94ac29aa/html5/thumbnails/16.jpg)
Examples of Neither• If f(x) = -3x³ + 2x², determine if f(x) is
even, odd, or neither.
1) Find f(-x).
2) f(-x) = -3(-x)³ + 2(-x)² = 3x³ + 2x²
3) Find –f(x).
4) -f(x) = 3x³ - 2x²
5) Because f(-x) ≠ f(x) and f(-x) ≠ -f(x), f(x) is neither even nor odd.