1. PreCalculus 1st Semester
1.3 Graphs of Functions
PreCalculus
1st Semester

2. Objectives:
Find domains and ranges of functions and use the Vertical Line Test for functions.
Determine intervals on which functions are increasing, decreasing , or constant.
Determine relative maximum and minimum values of functions.
Identify and graph step functions and other piecewise- defined functions.
Identify even and odd functions.

3. Interval Notation
Use intervals to write continuous, infinite sets with the following guidelines:
Brackets [] – indicates that the endpoint is included. Never use brackets with infinity.
Parenthesis () – indicates that the endpoint is not included.

4. The Graph of A Function
The graph of a function f is the collection of ordered pairs (x, f(x)) such that x is the domain of f.
x = the directed distance from the y-axis.
f(x) = the directed distance from the x-axis
To find domain: movement left to right
To find range: movement bottom to top

5. Example 1: Find the domain and range of the following functions:
(b)

6. Vertical Line Test
A set of points in a coordinate plane is the graph of y as a function of x if and only if no vertical line intersects the graph at more than one point.
Not a function
function

7. Increasing and Decreasing Functions
Trace the graph from left to right (x-axis) to indicate when the function increases, decreases or is constant.
Only look at the movement on the x-axis!
Always use parentheses for increasing and decreasing!
Decreasing (−∞, −3)
Constant (−3, 4)
Decreasing (4, ∞)

8. Example 2: (b) (a) Decreasing (−∞, 4) Decreasing (−∞, 0)
Increasing (0, 2)
Increasing (4, ∞)
Decreasing (2, ∞)

9. Example 2:
(c)
Decreasing (−∞, −1)
Constant (−1, 1)
Decreasing (1, ∞)

10. Minimum and Maximum Values
Points at which the function changes direction from increasing to decreasing (or decreasing to increasing)
These points are called critical points.
Relative Maximum – a high point on the visible graph.
Absolute Maximum – the highest point the graph ever reaches.
Relative Minimum – a low point on the visible graph.
Absolute Minimum – the lowest point the graph ever reaches.

11. Example 3:

12. Example 3:

13. Greatest Integer Function

14. Example 4: 1 2 3 -4 -3 2

15. Graphing Piecewise Functions
1)Rewrite in slope – intercept form
2) Graph using slope and y-intercept and/or choose points
3) Open dot < or >
4) Closed dot

16. Example 5: Graph

17. Even and Odd Functions
A function whose graph is symmetric with respect to the y-axis is an even function.
A function whose graph is symmetric with respect to the origin is an odd function.
A graph that is symmetric to the x-axis is not the graph of a function.

18. Even and Odd Functions
Three ways to determine if a function is even or odd:
Algebraically
Graphically
Examination

19. Algebraically:

20. Example 6:

21. Graphically:
Even
Even
Odd
Odd
Not a function

22. Examination: