1.2 (a) Functions and their properties…domain, range, continuity, discontinuity.

Definition: Function, Domain, and Range A function from a set D to a set R is a rule that assigns to every element in D a unique element in R. The set D of all input values is the domain of the function, and the set R of output values is the range of the function. Common notation: y = f(x) Here, x is the independent variable, and y is the dependent variable

VLT!!! (not a tasty BLT sandwich…) Vertical Line Test: A graph (set of points (x, y)) in the x-y plane defines y as a function of x if and only if no vertical line intersects the graph in more than one point.

Yup!!! Nope!!! Heck Naw!!! Yessir!!! Which of the following are graphs of functions? Yup!!! Nope!!! Heck Naw!!! Yessir!!!

Finding Domain and Range Agreement for Domain: Unless we are dealing with a model (like volume) that necessitates a restricted domain, we will assume that the domain of a function defined by an algebraic expression is the same as the domain of the algebraic expression, the implied domain. For models, we will use a domain that fits the situation, the relevant domain.

Finding Domain and Range Find the domain of the following functions (support graphically): The key question: Is there anything that x could not be??? Always write your answer in interval notation:

Finding Domain and Range Find the domain of the following functions (support graphically): What are the restrictions on x ? Interval notation:

Finding Domain and Range Find the range of the given function (use any method). What are the possible y-values for this function???

Finding Domain and Range Find the range of the given function (use any method). Check the graph…

The Concept of Continuity Algebraically, a function is continuous at x = a if Read “the limit of f (x) as x approaches a is f (a)” Graphically, a function is continuous at a particular point if the graph does not “come apart” at that point.  Let’s apply this with some examples…

How does that “limit definition” The Concept of Continuity Continuous at all x How does that “limit definition” apply???

Removable Discontinuity The Concept of Continuity Removable Discontinuity at x = a Why is it called “removable”? a

The Concept of Continuity Jump Discontinuity at x = a a

Infinite Discontinuity The Concept of Continuity Infinite Discontinuity at x = a a

Homework: p. 98 1-23 odd