1. Part 5: Even and Odd Functions

2. Even and Odd Functions
An even function has a graph which is symmetric with respect to the y –axis:
whenever (x, y) is on the graph, then so is the point
(–x, y).
An odd function has a graph which is symmetric with respect to the origin:
whenever (x, y) is on the graph, then so is the point (–x, –y).
Example: 𝑦= 𝑥 2
Example: 𝑓 𝑥 = 𝑥 3

3. Even and Odd Functions
A graph that is symmetric with respect to the x-axis is not the graph of a function (except for the graph of y = 0).
Symmetric to y-axis Even function
Symmetric to origin Odd function
Symmetric to x-axis Not a function

4. Test for Even and Odd Functions…
EVEN FUNCTION
ODD FUNCTION
𝒇 −𝒙 =𝒇(𝒙)
If you plug in a –x into the equation, then you get back the original equation, unchanged.
𝒇 −𝒙 =−𝒇(𝒙)
If you plug in a –x into the equation then you get back the negative of the original equation.
Example: 𝑓 𝑥 = 𝑥 2
𝑓 −𝑥 = −𝑥 2 = 𝑥 2
Example: 𝑓 𝑥 = 𝑥 3
𝑓 −𝑥 = −𝑥 3 = −𝑥 3

5. Example 10 – Even and Odd Functions
Determine whether each function is even, odd, or neither.
g(x) = x3 – x
b. h(x) = x2 + 1
c. f (x) = x3 – 1
d. k(x) = 2x + 3

6. Classwork…