1. Functions: Even/Odd/Neither
Math I: Unit 4 (Lesson 4)

2. Graphically… A function is even…
If the graph is symmetrical about the y-axis,
then it’s even **Fold hotdog!

3. Graphically… A function is odd…
If the graph is symmetrical about the y-axis &
x-axis (or symmetrical about the origin),
then it’s odd. **Fold hotdog & hamburger!

4. Examples: Graphically
Neither
Even
Odd

5. The equations are exactly the SAME…so EVEN function.
Algebraically…
A function is even if f(-x) = f(x)
Example 1: f(x) = 2x2 + 5
If you substitute in -x and get the SAME function that you started with, then it’s even.
f(-x)= f(x)
Replace x with –x
2x2 + 5
Simplify
The new equation
2(-x)2 + 5
2(-x)(-x) + 5
2x2 + 5
The equations are exactly the SAME…so EVEN function.

6. Algebraically… A function is odd if f(-x) = -f(x)
If you substitute in -x and get the OPPOSITE function
(all the signs change),then it’s odd.
Example: f(x) = 4x3 + 2x
f(-x) f(x)
Replace x with –x
4x3 + 2x
Simplify
The new equation
4(-x)3 + 2(-x)
4(-x)(-x)(-x) + 2(-x)
-4x3 – 2x
EVERY sign changed…so OPPOSITES…
ODD function

7. Neither… Graphically…
If a function does not have y-axis symmetry OR origin symmetry…then it has NEITHER.
Algebraically…
If, after substituting –x in place of x, the equation is not EXACTLY the same OR complete OPPOSITES, then the function is NEITHER.

8. Examples: Algebraically
f(x) = x4 + x2
f(x) = 1 + x3
f(x) = 2x3 + x
f(-x)
f(x)
(-x)4 + (-x)2
x4 + x2
(-x)(-x)(-x)(-x) + (-x)(-x)
f(-x)
f(x)
1 + (-x)3
1 + x3
1 + (-x)(-x)(-x)
1 – x3
f(-x)
f(x)
2(-x)3 + (-x)
2x3 + x
2(-x)(-x)(-x) + (-x)
2x3+ x
-2x3 - x
SAME – so EVEN
Not same and Not all signs changed – so NEITHER
OPPOSITES– so ODD