Odd and even functions
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The document defines even, odd, and neither functions based on their symmetry properties. An even function is symmetric about the y-axis, such that f(-x) = f(x). An odd function is symmetric about the origin, such that f(-x) = -f(x). A function is neither even nor odd if it does not satisfy those properties. Examples are provided to demonstrate how to determine if a given function is even, odd, or neither.
Odd and even functions
- 1. FUNCTIONS Symmetric about the y axis Symmetric about the origin
- 2. 2-7 -6 -5 -4 -3 -2 -1 1 5 730 4 6 8 7 1 2 3 4 5 6 8 -2 -3 -4 -5 -6 -7 So for an even function, for every point (x, y) on the graph, the point (-x, y) is also on the graph. Even functions have y-axis Symmetry
- 3. 2-7 -6 -5 -4 -3 -2 -1 1 5 730 4 6 8 7 1 2 3 4 5 6 8 -2 -3 -4 -5 -6 -7 So for an odd function, for every point (x, y) on the graph, the point (-x, -y) is also on the graph. Odd functions have origin Symmetry
- 4. 2-7 -6 -5 -4 -3 -2 -1 1 5 730 4 6 8 7 1 2 3 4 5 6 8 -2 -3 -4 -5 -6 -7 We wouldn’t talk about a function with x-axis symmetry because it wouldn’t BE a function. x-axis Symmetry
- 5. A function is even if f( -x) = f(x) for every number x in the domain. So if you plug a –x into the function and you get the original function back again it is even. ( ) 125 24 +−= xxxf Is this function even? ( ) 1251)(2)(5 2424 +−=+−−−=− xxxxxf YES ( ) xxxf −= 3 2 Is this function even? ( ) xxxxxf +−=−−−=− 33 2)()(2 NO
- 6. A function is odd if f( -x) = - f(x) for every number x in the domain. So if you plug a –x into the function and you get the negative of the function back again (all terms change signs) it is odd. ( ) 125 24 +−= xxxf Is this function odd? ( ) 1251)(2)(5 2424 +−=+−−−=− xxxxxf NO ( ) xxxf −= 3 2 Is this function odd? ( ) xxxxxf +−=−−−=− 33 2)()(2 YES
- 7. If a function is not even or odd we just say neither (meaning neither even nor odd) ( ) 15 3 −= xxf Determine if the following functions are even, odd or neither. ( ) ( ) 1515 33 −−=−−=− xxxf Not the original and all terms didn’t change signs, so NEITHER. ( ) 23 24 +−−= xxxf ( ) 232)()(3 2424 +−−=+−−−−=− xxxxxf Got f(x) back so EVEN.
- 8. Acknowledgement I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint. www.slcc.edu Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum. Stephen Corcoran Head of Mathematics St Stephen’s School – Carramar www.ststephens.wa.edu.au