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Even and Odd Functions
madelinemontavon.cmswiki.wikispaces.net
September 30th
What are even & odd functions?!
Plug 3 into this equation:
2x4 – 3x2 + 1 =
No plug -3 into this equation:
2x4 – 3x2 + 1 =
What do you notice about these answers?
What are even & odd functions?!
Plug 3 into this equation:
2x3 – 3x =
No plug -3 into this equation:
2x3 – 3x =
What do you notice about these answers?
What are even & odd functions?!
Plug 3 into this equation:
2x2 – 3x – 1 =
No plug -3 into this equation:
2x2 – 3x – 1 =
What do you notice about these answers?
Even Functions
A function is even when it is symmetric about the y – axis.
Meaning when you plug in (-x) the function simplifies to be the same:
f(-x) = f(x)Examples:
1. x2 – 1 2. 2x2 + 3 3. x4 + x2
Odd Functions
A function is odd when it is symmetric about the origin – the point (0, 0).
Meaning when you plug in (-x) and simplify the signs of each term changes:
f(-x) = -f(x)Examples:
1. x3 – x 2. x5 + x 3. 2x3
Functions that are NEITHER Odd nor Even A function is neither when it is NOT
symmetric about the y – axis or the origin.
Most functions are neither even nor odd
Meaning when you plug in (-x) you get a completely different function that is not the same and does not have all signs changed:
Examples:
1. x3 – x + 1 2. x4 + x 3. 2x3 + 5x – 4
Even Functions - Graphing
Even Functions – Graphing Determine which are even…
Odd Functions - Graphing
Odd Functions GraphingDetermine which are odd…
Even Functions-Algebraically
Plug in –x and simplify! If you get the same function, it is
even!
2424 52)(5)(2)( xxxxxf
24 52)( xxxf
Note: If the function has all EVEN exponents
and/or a constant (#) it will be even! That means it has
variable exponents of
2,4,6,etc…
Odd Functions-Algebraically
Plug in –x and simplify! If you get the opposite function, it is
odd!
233 52)(5)(2)( xxxxxf xxxf 52)( 3
Note: If the function has all ODD exponents & NO constant it will be odd! That means EVERY
variable term has an exponent of 1,3,5,7,etc…
Neither Functions-Algebraically
Plug in –x and simplify! If you do not get the same function,
or all of the signs do not change, it is neither!
525)(2)( 33 xxxf
52)( 3 xxf
If the function has a mixture of even or odd
exponents, or odd exponents & a constant then it will be neither!
This is the MOST COMMON!
Still -5
Example
Example
Example
Example
Example
Example
Determine if the function is even, odd, or neither.
42 73)( xxxf 43)( 2 xxf
Example
Determine if the function is even, odd, or neither.
xxxf 42)( 3 542)( 23 xxxf
Example
Determine if the function is even, odd, or neither.
25)( xxxf xxxf 54)( 3
Example
Determine if the function is even, odd, or neither.
5
35)(
x
xxf
xx
xxxf
5
4
)(
NC Final Exam Question Example
Which of the following is an even function?
8)6(4)
34)
624)
134)
2
2
2
2
xd
xc
xxb
xxa
Challenge Question
1. Is (x+1)2 even, odd, or neither?
2. Can a function be both even and odd?
Classwork
Complete the front of the worksheet with a partner
Homework Complete Worksheet Need more help? Check out this website:
http://www.mathsisfun.com/algebra/functions-odd-even.html
https://www.khanacademy.org/math/algebra2/functions_and_graphs/analyzing_functions/v/recognizing-odd-and-even-functions
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