Relation A relation is a correspondence between two sets where each element in the first set, called...
-
Upload
leon-williamson -
Category
Documents
-
view
218 -
download
0
Transcript of Relation A relation is a correspondence between two sets where each element in the first set, called...
![Page 1: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/1.jpg)
![Page 2: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/2.jpg)
Relation
A relation is a correspondence
between two sets where each element
in the first set, called the domain, corresponds to at least one element in the
second set, called the range.
![Page 3: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/3.jpg)
Relation
Person Blood type
ORDERED PAIR
Michael A (Michael, A)
Tania A (Tania, A)
Dylan AB (Dylan, AB)
Trevor 0 (Trevor, O)
Megan 0 (Megan, O)
![Page 4: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/4.jpg)
Relation
Person Blood type
ORDERED PAIR
Michael A (Michael, A)
Tania A (Tania, A)
Dylan AB (Dylan, AB)
Trevor 0 (Trevor, O)
Megan 0 (Megan, O)
![Page 5: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/5.jpg)
Relation
Person Blood type
ORDERED PAIR
Michael A (Michael, A)
Tania A (Tania, A)
Dylan AB (Dylan, AB)
Trevor 0 (Trevor, O)
Megan 0 (Megan, O)
The domain is the set of all the first components.
{Michael, Tania, Dylan, Trevor, Megan}
The range is the set of all the second components.
{A, AB, O}
![Page 6: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/6.jpg)
Relation
Person Blood type
ORDERED PAIR
Michael A (Michael, A)
Tania A (Tania, A)
Dylan AB (Dylan, AB)
Trevor 0 (Trevor, O)
Megan 0 (Megan, O)
The domain is the set of all the first components.
{Michael, Tania, Dylan, Trevor, Megan}
The range is the set of all the second components.
{A, AB, O}
![Page 7: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/7.jpg)
Function
![Page 8: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/8.jpg)
Function
• A function is a correspondence between two sets where each element in the first set, called the domain, corresponds to exactly one element in the second set, called the range
![Page 9: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/9.jpg)
Function
• A function is a correspondence between two sets where each element in the first set, called the domain, corresponds to exactly one element in the second set, called the range
• Note that the definition of a function is more restrictive than the definition of a relation.
![Page 10: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/10.jpg)
Function
Time of day Competition
1:00 P.M. Football
2:00 P.M. Volleyball
7:00 P.M. Soccer
7:00 P.M. Basketball
![Page 11: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/11.jpg)
Functions Defined by Equations
![Page 12: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/12.jpg)
Functions Defined by Equations
y = x2 − 3x
![Page 13: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/13.jpg)
Functions Defined by Equations
y = x2 − 3x
x y = x2 − 3x y
![Page 14: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/14.jpg)
Functions Defined by Equations
y = x2 − 3x
x y = x2 − 3x y
1
![Page 15: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/15.jpg)
Functions Defined by Equations
y = x2 − 3x
x y = x2 − 3x y
1 y = (1)2 − 3(1)
![Page 16: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/16.jpg)
Functions Defined by Equations
y = x2 − 3x
x y = x2 − 3x y
1 y = (1)2 − 3(1) −2
![Page 17: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/17.jpg)
Functions Defined by Equations
y = x2 − 3x
x y = x2 − 3x y
1 y = (1)2 − 3(1) −2
5
![Page 18: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/18.jpg)
Functions Defined by Equations
y = x2 − 3x
x y = x2 − 3x y
1 y = (1)2 − 3(1) −2
5 y = (5)2 − 3(5)
![Page 19: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/19.jpg)
Functions Defined by Equations
y = x2 − 3x
x y = x2 − 3x y
1 y = (1)2 − 3(1) −2
5 y = (5)2 − 3(5) 10
![Page 20: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/20.jpg)
Functions Defined by Equations
y = x2 − 3x
x y = x2 − 3x y
1 y = (1)2 − 3(1) −2
5 y = (5)2 − 3(5) 10
1.2
![Page 21: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/21.jpg)
Functions Defined by Equations
y = x2 − 3x
x y = x2 − 3x y
1 y = (1)2 − 3(1) −2
5 y = (5)2 − 3(5) 10
1.2 y = (1.2)2 − 3(1.2)
![Page 22: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/22.jpg)
Functions Defined by Equations
y = x2 − 3x
x y = x2 − 3x y
1 y = (1)2 − 3(1) −2
5 y = (5)2 − 3(5) 10
1.2 y = (1.2)2 − 3(1.2) −2.16
![Page 23: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/23.jpg)
Functions Defined by Equations
y = x2 − 3x
x y = x2 − 3x y
• Since the variable y depends on what value of x is selected, we denote y as
the dependent variable. (output)
![Page 24: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/24.jpg)
Functions Defined by Equations
y = x2 − 3x
x y = x2 − 3x y
• Since the variable y depends on what value of x is selected, we denote y as
the dependent variable. (output)
• The variable x can be any number in the domain; therefore, we denote x as the independent variable. (input)
![Page 25: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/25.jpg)
Function Notation
![Page 26: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/26.jpg)
Function Notation
• The notation y = f(x) denotes that the variable y is function of x.
![Page 27: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/27.jpg)
Function Notation
• The notation y = f(x) denotes that the variable y is function of x.
INPUT FUNCTION OUTPUT EQUATION
x f f (x) f (x) = 2x + 5
![Page 28: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/28.jpg)
Function Notation
• A Linear function is a function defined by an equation that can be written in the form
f(x) = mx + b , or y = mx + b
where m is the slope of the line graph and
(0, b) is the y - intercept
![Page 29: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/29.jpg)
Function Notation
• A Linear function is a function defined by an equation that can be written in the form
f(x) = mx + b , or y = mx + b
where m is the slope of the line graph and
(0, b) is the y - intercept
Ex. y = -3x + 8 f(x) = 5x – 4
![Page 30: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/30.jpg)
The Graph of the Function
• The graph of the function is the graph of the ordered pairs (x, f(x)), that define the function.
![Page 31: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/31.jpg)
Use the given graphs to evaluate the function.
Find f (0), f (1). f (2) , 4f (3),
Find x such that f (x) = 10, f (x) = 2
Find x such that f (x) = 10, f (x) = 2
![Page 32: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/32.jpg)
Use the given graphs to evaluate the function.
T(−5)
T(−2)
T(4)
![Page 33: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/33.jpg)
Vertical Line Test
• Given the graph of an equation, if any vertical line that can be drawn intersects the graph at no more than one point, the
equation defines a function of x.
This test is called the vertical line test.
![Page 34: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/34.jpg)
Vertical Line Test
![Page 35: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/35.jpg)
Evaluating the Difference Quotient
![Page 36: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/36.jpg)
Evaluating the Difference Quotient
( ) ( )f x h f x
h
![Page 37: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/37.jpg)
Evaluating the Difference Quotient
For the function f (x) = x2 − x, find
( ) ( )f x h f x
h
![Page 38: Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in.](https://reader036.fdocuments.in/reader036/viewer/2022062400/56649eaa5503460f94baf7a4/html5/thumbnails/38.jpg)
Falling Objects: Firecrackers.
• A firecracker is launched straight up, and its height is a function of time,
h(t) = −16t2 + 128t, where h is the height in feet and t is the time in seconds with
t = 0 corresponding to the instant it launches. What is the height 4 seconds after launch?