1. Properties of Functions
Dr. Fowler  AFM  Unit 1-3
Properties of Functions

3. Even and Odd Functions (graphically)
If the graph of a function is symmetric with respect to the y-axis, then it’s even.
If the graph of a function is symmetric with respect to the origin, then it’s odd.
The easiest thing to do is to plug in 1 and -1 (or 2 and -2)
if you get the same y, then it’s Even.
If you get the opposite y, then it’s Odd.
If you get different y’s, then it’s Neither.
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4. Determine whether each graph given is an even function, an odd function, or a function that is neither even nor odd.
Even function because it is symmetric with respect to the y-axis
Neither even nor odd because no symmetry with respect to the y-axis or the origin.
Odd function because it is symmetric with respect to the origin.

5. VIDEO - Odd, Even, or Neither Function?

6. Even and Odd Functions (algebraically)
A function is even if f(-x) = f(x)
If you plug in -x and get the original function, then it’s even.
The easiest thing to do is to plug in 1 and -1 (or 2 and -2)
if you get the same y, then it’s Even.
If you get the opposite y, then it’s Odd.
If you get different y’s, then it’s Neither.
A function is odd if f(-x) = -f(x)
If you plug in -x and get the opposite function, then it’s odd.
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7. EVEN Example 1 Even, Odd or Neither? Graphically Algebraically
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8. ODD Example 2 They are opposite, so… Even, Odd or Neither? Graphically
What happens if we plug in 2?
Graphically
Algebraically
ODD
They are opposite, so…
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9. Neither Example 4 Even, Odd or Neither? Graphically Algebraically
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11. Where is the function increasing?
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12. Where is the function decreasing?
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13. Where is the function constant?
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15. The local maximum value is 2. Y values
Local maximum when x = 1. Before a turn back down.
The local maximum value is 2. Y values

16. The local minima values are 1 and 0. (y values)
Local minimum when x = –1 and x = 3. X values before a turn up.
The local minima values are 1 and 0. (y values)

17. (e) List the intervals on which f is increasing.
(f) List the intervals on which f is decreasing.
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18. Find the absolute maximum and the absolute minimum, if they exist.
The absolute maximum of 6 occurs when x = 3.
The absolute minimum of 1 occurs when x = 0.
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19. Find the absolute maximum and the absolute minimum, if they exist.
The absolute maximum of 4 occurs when x = 5.
The absolute minimum of 1 occurs on the interval [1,2].
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20. Find the absolute maximum and the absolute minimum, if they exist.
There is no absolute maximum.
The absolute minimum of 0 occurs when x = 0.
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22. a) From 1 to 3
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23. b) From 1 to 5
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24. c) From 1 to 7
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25. Excellent Job !!! Well Done

26. Stop Notes for Today.
Do Worksheet

27. For an even function, for every point (x, y) on the graph, the point (-x, y) is also on the graph.
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28. So for an odd function, for every point (x, y) on the graph, the point (-x, -y) is also on the graph.
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29. Odd function symmetric with respect to the origin
Even function symmetric with respect to the y-axis
Since the resulting function does not equal f(x) nor –f(x) this function is neither even nor odd and is not symmetric with respect to the y-axis or the origin.
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35. Find the absolute maximum and the absolute minimum, if they exist.
The absolute maximum of 3 occurs when x = 5.
There is no absolute minimum because of the “hole” at x = 3.
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36. Find the absolute maximum and the absolute minimum, if they exist.
There is no absolute maximum.
There is no absolute minimum.
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42. -4 3 2
-25
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43. Warmup Which of the following are graphs of functions? A Function
Not a Function
Not a Function

44. EVEN Example 3 Even, Odd or Neither? Graphically Algebraically
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