1. Graphs of Functions
Lesson 3

2. Warm Up – Perform the Operations and Simplify

3. Solution

4. Solution

5. Solution

6. Solution

7. Domain & Range of a Function
What is the domain of
the graph of the function f?

8. Domain & Range of a Function
What is the range of
the graph of the function f?

9. Domain & Range of a Function

10. Let’s look at domain and range of a function using an algebraic approach.
Then, let’s check it with a graphical approach.

11. Find the domain and range of
Algebraic Approach
The expression under the radical can not be negative.
Therefore, Domain
Since the domain is never negative the range is the set of all nonnegative real
numbers.

12. Find the domain and range of
Graphical Approach

13. Increasing and Decreasing Functions

14. The more you know about the graph of a function, the more you know about the function itself.
Consider the graph on the next slide.

15. Falls from x = -2 to x = 0.
Is constant from x = 0 to
x = 2.
Rises from x = 2 to x = 4.

16. Increases over the entire real line.
Ex: Find the open intervals on which the function is increasing, decreasing, or constant.
Increases over the entire real line.

17. Ex: Find the open intervals on which the function is increasing, decreasing, or constant.

18. Ex: Find the open intervals on which the function is increasing, decreasing, or constant.

19. Relative Minimum and Maximum Values

20. The point at which a function changes its increasing, decreasing, or constant behavior are helpful in determining the relative maximum or relative minimum values of a function.

22. General Points – We’ll find EXACT points later……

23. Approximating a Relative Minimum
Example: Use a GDC to approximate the relative minimum of the function given by

24. Put the function into the “y = “ the press zoom 6 to look at the graph.
Press trace to follow the line to the lowest point.

25. Example
Use a GDC to approximate the relative minimum and relative maximum of the function given by

26. Solution
Relative Minimum
(-0.58, -0.38)

27. Solution
Relative Maximum
(0.58, 0.38)

28. Step Functions and Piecewise-Defined Functions

29. Because of the vertical jumps, the greatest integer function is an example
of a step function.

30. Let’s graph a Piecewise-Defined Function
Sketch the graph of
Notice when open dots and closed
dots are used. Why?

31. Even and Odd Functions

32. Graphically

33. Algebraically
Let’s look at the graphs again and see if this applies.

34. Graphically ☺ ☺

35. Example
Determine whether each function is even, odd, or neither.

36. Graphical – Symmetric to Origin
Algebraic

37. Algebraic
Graphical – Symmetric to y-axis

38. Graphical – NOT Symmetric to origin OR y-axis.
Algebraic

39. You Try
Is the function
Even, Odd, of Neither?

40. Solution
Symmetric about the y-axis.