Do Now from Do Now from 1.1b…1.1b…

Solve the equation graphically by Solve the equation graphically by converting it into an equivalent converting it into an equivalent equation with 0 on the right-hand side equation with 0 on the right-hand side and then finding the x-interceptsand then finding the x-intercepts3 2 2 ( 8)x x

( 6) 6 2 (5 )x x

1.09,2.86x

2.66x

Functions and their Functions and their properties…domain, properties…domain,

range, continuity, range, continuity, discontinuity.discontinuity.

We are functioning wellWe are functioning well

in Sec. 1.2a!!!in Sec. 1.2a!!!

Definition: Function, Domain, and RangeDefinition: Function, Domain, and Range

A function from a set D to a set R is a rule thatassigns to every element in D a unique element in R.The set D of all input values is the domain of thefunction, and the set R of output values is the rangeof the function.

Common notation: y = f(x)

Here, x is the independent variable,and y is the dependent variable

VLT!!!VLT!!! (not a tasty BLT sandwich…) (not a tasty BLT sandwich…)

Vertical Line Test: A graph (set of points (x, y)) inthe x-y plane defines y as a function of x if and onlyif no vertical line intersects the graph in more thanone point.

Which of the following are graphs of functions?

Yup!!!Yup!!!Nope!!!Nope!!!

Yessir!!!Yessir!!! Heck Naw!!!Heck Naw!!!

Finding Domain and RangeFinding Domain and RangeAgreement for Domain: Unless we are dealing witha model (like volume) that necessitates a restricteddomain, we will assume that the domain of a functiondefined by an algebraic expression is the same as thedomain of the algebraic expression, the implieddomain. For models, we will use a domain that fitsthe situation, the relevant domain.

Finding Domain and RangeFinding Domain and RangeFind the domain of the following functions (support graphically):

3f x x

D : 3,

The key question: Is there anything that x could not be???

3 0x 3x

Always write your answerin interval notation:

Finding Domain and RangeFinding Domain and RangeFind the domain of the following functions (support graphically):

5

xg x

x

D : 0,5 5,

What are the restrictions on x ?

5 0x 5x

0x Interval notation:

Finding Domain and RangeFinding Domain and RangeFind the range of the given function (use any method).

5 4g x x

R : 5,What are the possible y-values for this function???

Finding Domain and RangeFinding Domain and RangeFind the range of the given function (use any method).

2

2

3

4

xg x

x

R : , 1 3 4,

Check the graph…

The Concept of The Concept of ContinuityContinuityAlgebraically, a function is continuous at x = a if

limx a

f x f a

Graphically, a function is continuous at a particular point if thegraph does not “come apart” at that point.

Let’s apply this with some examples…Let’s apply this with some examples…

Read “the limit of f (x) as x approaches a is f (a)”

The Concept of The Concept of ContinuityContinuity

Continuous at all xContinuous at all x

How does that “limit definition”

apply???

The Concept of The Concept of ContinuityContinuity

Removable DiscontinuityRemovable Discontinuityat x = aat x = a

a Why is it called“removable”?

The Concept of The Concept of ContinuityContinuity

Jump DiscontinuityJump Discontinuityat x = aat x = a

a

The Concept of The Concept of ContinuityContinuity

Infinite DiscontinuityInfinite Discontinuityat x = aat x = a

a

Homework: p. 98 1-23 odd