Functions and their properties…domain, range, continuity, discontinuity. We are functioning well...
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Transcript of Functions and their properties…domain, range, continuity, discontinuity. We are functioning well...
![Page 1: Functions and their properties…domain, range, continuity, discontinuity. We are functioning well in Sec. 1.2a!!! Homework: p. 98 1-23 odd.](https://reader030.fdocuments.net/reader030/viewer/2022032709/56649eaa5503460f94baf7bc/html5/thumbnails/1.jpg)
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!!!
Homework: p. 98 1-23 Homework: p. 98 1-23 oddodd
![Page 2: Functions and their properties…domain, range, continuity, discontinuity. We are functioning well in Sec. 1.2a!!! Homework: p. 98 1-23 odd.](https://reader030.fdocuments.net/reader030/viewer/2022032709/56649eaa5503460f94baf7bc/html5/thumbnails/2.jpg)
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
![Page 3: Functions and their properties…domain, range, continuity, discontinuity. We are functioning well in Sec. 1.2a!!! Homework: p. 98 1-23 odd.](https://reader030.fdocuments.net/reader030/viewer/2022032709/56649eaa5503460f94baf7bc/html5/thumbnails/3.jpg)
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.
![Page 4: Functions and their properties…domain, range, continuity, discontinuity. We are functioning well in Sec. 1.2a!!! Homework: p. 98 1-23 odd.](https://reader030.fdocuments.net/reader030/viewer/2022032709/56649eaa5503460f94baf7bc/html5/thumbnails/4.jpg)
Which of the following are graphs of functions?
Yup!!!Yup!!!Nope!!!Nope!!!
Yessir!!!Yessir!!! Heck Naw!!!Heck Naw!!!
![Page 5: Functions and their properties…domain, range, continuity, discontinuity. We are functioning well in Sec. 1.2a!!! Homework: p. 98 1-23 odd.](https://reader030.fdocuments.net/reader030/viewer/2022032709/56649eaa5503460f94baf7bc/html5/thumbnails/5.jpg)
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.
![Page 6: Functions and their properties…domain, range, continuity, discontinuity. We are functioning well in Sec. 1.2a!!! Homework: p. 98 1-23 odd.](https://reader030.fdocuments.net/reader030/viewer/2022032709/56649eaa5503460f94baf7bc/html5/thumbnails/6.jpg)
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:
![Page 7: Functions and their properties…domain, range, continuity, discontinuity. We are functioning well in Sec. 1.2a!!! Homework: p. 98 1-23 odd.](https://reader030.fdocuments.net/reader030/viewer/2022032709/56649eaa5503460f94baf7bc/html5/thumbnails/7.jpg)
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:
![Page 8: Functions and their properties…domain, range, continuity, discontinuity. We are functioning well in Sec. 1.2a!!! Homework: p. 98 1-23 odd.](https://reader030.fdocuments.net/reader030/viewer/2022032709/56649eaa5503460f94baf7bc/html5/thumbnails/8.jpg)
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???
![Page 9: Functions and their properties…domain, range, continuity, discontinuity. We are functioning well in Sec. 1.2a!!! Homework: p. 98 1-23 odd.](https://reader030.fdocuments.net/reader030/viewer/2022032709/56649eaa5503460f94baf7bc/html5/thumbnails/9.jpg)
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…
![Page 10: Functions and their properties…domain, range, continuity, discontinuity. We are functioning well in Sec. 1.2a!!! Homework: p. 98 1-23 odd.](https://reader030.fdocuments.net/reader030/viewer/2022032709/56649eaa5503460f94baf7bc/html5/thumbnails/10.jpg)
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)”
![Page 11: Functions and their properties…domain, range, continuity, discontinuity. We are functioning well in Sec. 1.2a!!! Homework: p. 98 1-23 odd.](https://reader030.fdocuments.net/reader030/viewer/2022032709/56649eaa5503460f94baf7bc/html5/thumbnails/11.jpg)
The Concept of The Concept of ContinuityContinuity
Continuous at all xContinuous at all x
How does that “limit definition”
apply???
![Page 12: Functions and their properties…domain, range, continuity, discontinuity. We are functioning well in Sec. 1.2a!!! Homework: p. 98 1-23 odd.](https://reader030.fdocuments.net/reader030/viewer/2022032709/56649eaa5503460f94baf7bc/html5/thumbnails/12.jpg)
The Concept of The Concept of ContinuityContinuity
Removable DiscontinuityRemovable Discontinuityat x = aat x = a
a Why is it called“removable”?
![Page 13: Functions and their properties…domain, range, continuity, discontinuity. We are functioning well in Sec. 1.2a!!! Homework: p. 98 1-23 odd.](https://reader030.fdocuments.net/reader030/viewer/2022032709/56649eaa5503460f94baf7bc/html5/thumbnails/13.jpg)
The Concept of The Concept of ContinuityContinuity
Jump DiscontinuityJump Discontinuityat x = aat x = a
a
![Page 14: Functions and their properties…domain, range, continuity, discontinuity. We are functioning well in Sec. 1.2a!!! Homework: p. 98 1-23 odd.](https://reader030.fdocuments.net/reader030/viewer/2022032709/56649eaa5503460f94baf7bc/html5/thumbnails/14.jpg)
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