Period 1: Odd and Even Functions

Even Functions

- a function that can be reflected in the y axis and remain unchanged (symmetric about the y axis)

Algebraically to find if a function is odd, one must sub f(-x) into the function

For ex: f(x) = x^2 - 3
f(-x) = (-x)^2 - 3 (watch for negatives since (-x)^2 = (-x)(-x))
f(-x) = x^2 - 3 = f(x)

Because f(x) is the same as f(-x) then it's an even function.

Odd Functions

- a function that can be reflected first in the y axis then in the x axis and remain unchanged (basically rotating at the origin for 180 degrees)

f(x) = x^3 is a typical odd function

To find out if a function is odd algebraically, one subs -f(-x) into an equation:

f(x) = 2x^3
f(-x) = 2(-x)^3
f(-x) = -2x^3 which does not equal f(x) so it's not even
-f(-x) = -1[-2x^3]
-f(-x) = 2x^3 = f(x) therefore since f(x) equals -f(-x) then the function is odd

It is also possible for a function to be neither even or odd.

Quick Check

Even function: if every term in the function has a degree with an even number then the function is even

Ex: f(x) = x^6 + x^4 + 3 (3 = x^0)

Odd function: if every term in the function has a degree with an odd number then the function is odd

Ex: f(x) = x^5 + x