Functions and Graphs Chapter 1, Section 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)

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

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

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

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

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.

Point discontinuity - Examples

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

Jump discontinuity - examples

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

Infinite discontinuity - examples

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.

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: