Even and odd functions.

1. OK, Really this
time, Even and
uh ... ODD (?)
Functions
~ BLINK ~ by flickr user ViaMoi

2. Given A(-2, -3) find the coordinates of its image under the
transformation given above.
The image of point B after the transformation shown above is (1, 4).
Find the original coordinates of B.

3. EVEN FUNCTIONS
Graphically: A function is quot;evenquot; if its graph is symmetrical about the y-axis.
These functions
are even...
These are
not ...
Symbolically (Algebraically)
a function is quot;evenquot; IFF (if and only if) ƒ(-x) = ƒ(x)
Examples: Are these functions even?
1. f(x) = x² 2. g(x) = x² + 2x
f(-x) = (-x)² g(-x) = (-x)² + 2(-x)
f(-x) = x² g(-x) = x² - 2x
since f(-x)=f(x) since g(-x) is not equal to g(x)
f is an even function g is not an even function

4. ODD FUNCTIONS
Graphically: A function is quot;oddquot; if its graph is symmetrical about the origin.
These
functions
These are
are odd ... not ...
Symbolically (Algebraically)
a function is quot;oddquot; IFF (if and only if) ƒ(-x) = -ƒ(x)
1. ƒ(x) = x³ - x 2. g(x) = x³- x²
Examples: ƒ(-x) = (-x)³ - (-x) g(-x) = (-x)³ - (-x)²
ƒ(x) = -x³ + x g(x) = -x³ - x²
-ƒ(x) = -(x³ - x) -g(x) = -(x³-x²)
-ƒ(x) = -x³ + x -g(x) = -x³+ x²
since ƒ(-x)= -ƒ(x) since g(-x) is not equal to -g(x)
ƒ is an odd function g is not an odd function

5. Are these functions even or odd? Justify your answers algebraically.
3
g(x) = x + 3x
4
ƒ(x) = x + 2x + 3 2