1. Functions and Graphs
Chapter 1, Section 2

2. Functions
Definition: a rule that assigns to every element in the domain a unique element in the range
Notation: f(x)
Passes vertical line test
Domain: set of all x-values (also called independent values)
Range: set of all y-values (also called dependent values)

3. Functions examples Is it a function? Domain? Range? Is it a function?

4. Functions examples Is it a function? Domain? Range? Is it a function?

5. Types of Functions Continuous Functions Discontinuous Functions
Graph does not “come apart”
Discontinuous Functions
Graph does “come apart”

6. Types of Discontinuity
Point discontinuity (removable)
Jump discontinuity
Infinite discontinuity

7. Point discontinuity There is a “hole” in the graph
The graph appears normal, the table shows error
To find algebraically, find the value(s) that make the denominator AND numerator = 0
This is considered a “removable” discontinuity because the function can be simplified to “remove” the value that causes the hole.

8. Point discontinuity - Examples

9. Jump discontinuity There is a jump in the graph at a specific x-value
Generally seen as piecewise functions
**consider the domain restrictions first on piecewise functions**
Can NOT be considered removable because there is no way to simplify the function

10. Jump discontinuity - examples

11. Infinite discontinuity
There are “broken” pieces of the graph
Will see the discontinuity on the graph and error on the table
To find algebraically, find the value(s) that will make the denominator = 0
**the denominator always has a higher degree than the numerator**
These are the functions that have vertical asymptotes

12. Infinite discontinuity - examples

13. Calculus and Continuous Functions
Continuous functions are main points for studying the concept of limits in calculus. This means that a function is continuous at some value a if the limit of that function as x gets closer to a exists.

14. In Conclusion
Exit Slip: Create a summary of the information you learned about continuity. Include an example for each type (not one from your notes) Homework: