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Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Section 2.3
Properties of Functions
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
For an even function, for every point (x, y) on the
graph, the point (-x, y) is also on the graph.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
So for an odd function, for every point (x, y) on the
graph, the point (-x, -y) is also on the graph.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
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
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
( ) 3
) 5a f x x x= + ( ) ( ) ( )
3
5f x x x− = − + − 3
5x x= − −
( ) ( )3
5x x f x= − + = −Odd function symmetric with
respect to the origin
( ) 2
) 2 3b g x x= − ( ) ( )
2
32g x x− = − − = 2x2
− 3= g(x)
Even function symmetric
with respect to the y-axis
( ) 3
) 14c h x x= − + ( ) ( )
3
4 1h x x− = − − + 3
4 1x= +
Since the resulting function does not equal h(x) nor –h(x) this function is
neither even nor odd and is not symmetric with respect to the y-axis or the
origin.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
IN
C
R
EA
SIN
G
DECR
EASIN
G
CONSTANT
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Where is the function
increasing?
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Where is the function
decreasing?
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Where is the function
constant?
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
There is a local maximum when x = 1.
The local maximum value is 2.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
There is a local minimum when x = –1 and x = 3.
The local minima values are 1 and 0.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
(e) List the intervals on which f is increasing.
(f) List the intervals on which f is decreasing.
( ) ( )1,1 and 3,− ∞
( ) ( ), 1 and 1,3−∞ −
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
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.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
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.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
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].
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
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.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Find the absolute maximum and the absolute minimum, if they exist.
There is no absolute maximum.
There is no absolute minimum.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
a) From 1 to 3
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
b) From 1 to 5
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
c) From 1 to 7
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
( ) 2
Suppose that 2 4 3.g x x x= − + −
( ) ( )( )
( )
22
2(1) 4(1) 3 2 2 4 2 3 18
(a) 6
1 2 3
y
x
− + − − − − + − −∆
= = =
∆ − −
( ) ( 19) 6( ( 2))b y x− − = − −
19 6 12y x+ = +
6 7y x= −
-4 3
2
-25

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Section 2.3 properties of functions

  • 1. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Section 2.3 Properties of Functions
  • 2. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 3. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. For an even function, for every point (x, y) on the graph, the point (-x, y) is also on the graph.
  • 4. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. So for an odd function, for every point (x, y) on the graph, the point (-x, -y) is also on the graph.
  • 5. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 6. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 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
  • 7. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 8. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. ( ) 3 ) 5a f x x x= + ( ) ( ) ( ) 3 5f x x x− = − + − 3 5x x= − − ( ) ( )3 5x x f x= − + = −Odd function symmetric with respect to the origin ( ) 2 ) 2 3b g x x= − ( ) ( ) 2 32g x x− = − − = 2x2 − 3= g(x) Even function symmetric with respect to the y-axis ( ) 3 ) 14c h x x= − + ( ) ( ) 3 4 1h x x− = − − + 3 4 1x= + Since the resulting function does not equal h(x) nor –h(x) this function is neither even nor odd and is not symmetric with respect to the y-axis or the origin.
  • 9. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. IN C R EA SIN G DECR EASIN G CONSTANT
  • 10. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Where is the function increasing?
  • 11. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Where is the function decreasing?
  • 12. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Where is the function constant?
  • 13. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 14. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 15. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 16. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 17. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 18. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 19. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. There is a local maximum when x = 1. The local maximum value is 2.
  • 20. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. There is a local minimum when x = –1 and x = 3. The local minima values are 1 and 0.
  • 21. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. (e) List the intervals on which f is increasing. (f) List the intervals on which f is decreasing. ( ) ( )1,1 and 3,− ∞ ( ) ( ), 1 and 1,3−∞ −
  • 22. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 23. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 24. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 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.
  • 25. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 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.
  • 26. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 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].
  • 27. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 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.
  • 28. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Find the absolute maximum and the absolute minimum, if they exist. There is no absolute maximum. There is no absolute minimum.
  • 29. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 30. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 31. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 32. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. a) From 1 to 3
  • 33. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. b) From 1 to 5
  • 34. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. c) From 1 to 7
  • 35. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 36. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 37. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. ( ) 2 Suppose that 2 4 3.g x x x= − + − ( ) ( )( ) ( ) 22 2(1) 4(1) 3 2 2 4 2 3 18 (a) 6 1 2 3 y x − + − − − − + − −∆ = = = ∆ − − ( ) ( 19) 6( ( 2))b y x− − = − − 19 6 12y x+ = + 6 7y x= − -4 3 2 -25