do now from 1.1b… solve the equation graphically by converting it into an equivalent equation with...
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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
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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!!!
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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
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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.
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Which of the following are graphs of functions?
Yup!!!Yup!!!Nope!!!Nope!!!
Yessir!!!Yessir!!! Heck Naw!!!Heck Naw!!!
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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.
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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:
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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:
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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???
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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…
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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)”
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The Concept of The Concept of ContinuityContinuity
Continuous at all xContinuous at all x
How does that “limit definition”
apply???
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The Concept of The Concept of ContinuityContinuity
Removable DiscontinuityRemovable Discontinuityat x = aat x = a
a Why is it called“removable”?
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The Concept of The Concept of ContinuityContinuity
Jump DiscontinuityJump Discontinuityat x = aat x = a
a
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The Concept of The Concept of ContinuityContinuity
Infinite DiscontinuityInfinite Discontinuityat x = aat x = a
a
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Homework: p. 98 1-23 odd