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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
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