Relations and Functions. Review A relation between two variables x and y is a set of ordered pairs...
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Transcript of Relations and Functions. Review A relation between two variables x and y is a set of ordered pairs...
![Page 1: Relations and Functions. Review A relation between two variables x and y is a set of ordered pairs An ordered pair consist of a x and y-coordinate A relation.](https://reader035.fdocuments.us/reader035/viewer/2022070410/56649f1f5503460f94c36dc4/html5/thumbnails/1.jpg)
Relations and Functions
![Page 2: Relations and Functions. Review A relation between two variables x and y is a set of ordered pairs An ordered pair consist of a x and y-coordinate A relation.](https://reader035.fdocuments.us/reader035/viewer/2022070410/56649f1f5503460f94c36dc4/html5/thumbnails/2.jpg)
Review A relation between two variables x and y
is a set of ordered pairs An ordered pair consist of a x and y-
coordinate A relation may be viewed as ordered pairs,
mapping design, table, equation, or written in sentences
x-values are inputs, domain, independent variable
y-values are outputs, range, dependent variable
![Page 3: Relations and Functions. Review A relation between two variables x and y is a set of ordered pairs An ordered pair consist of a x and y-coordinate A relation.](https://reader035.fdocuments.us/reader035/viewer/2022070410/56649f1f5503460f94c36dc4/html5/thumbnails/3.jpg)
Example 1
{( , ), ( , ), ( , ), ( , ), ( , ), ( , )} 0 5 1 4 2 3 3 2 4 1 5 0
•What is the domain? {0, 1, 2, 3, 4, 5}What is the range?
{-5, -4, -3, -2, -1, 0}
![Page 4: Relations and Functions. Review A relation between two variables x and y is a set of ordered pairs An ordered pair consist of a x and y-coordinate A relation.](https://reader035.fdocuments.us/reader035/viewer/2022070410/56649f1f5503460f94c36dc4/html5/thumbnails/4.jpg)
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Example 2
•What is the domain? {4, -5, 0, 9, -1}
•What is the range?{-2, 7}
Input 4 –5 0 9 –1
–2 7Output
![Page 5: Relations and Functions. Review A relation between two variables x and y is a set of ordered pairs An ordered pair consist of a x and y-coordinate A relation.](https://reader035.fdocuments.us/reader035/viewer/2022070410/56649f1f5503460f94c36dc4/html5/thumbnails/5.jpg)
Is a relation a function?
What is a function?
According to the textbook, “a function is…a relation in
which every input is paired with exactly one output”
![Page 6: Relations and Functions. Review A relation between two variables x and y is a set of ordered pairs An ordered pair consist of a x and y-coordinate A relation.](https://reader035.fdocuments.us/reader035/viewer/2022070410/56649f1f5503460f94c36dc4/html5/thumbnails/6.jpg)
Is a relation a function?•Focus on the x-coordinates, when given a relation
If the set of ordered pairs have different x-coordinates, it IS A function
If the set of ordered pairs have same x-coordinates, it is NOT a function
•Y-coordinates have no bearing in determining functions
![Page 7: Relations and Functions. Review A relation between two variables x and y is a set of ordered pairs An ordered pair consist of a x and y-coordinate A relation.](https://reader035.fdocuments.us/reader035/viewer/2022070410/56649f1f5503460f94c36dc4/html5/thumbnails/7.jpg)
Example 3
YES
{( , ), ( , ), ( , ), ( , ), ( , ), ( , )} 0 5 1 4 2 3 3 2 4 1 5 0
•Is this a function?•Hint: Look only at the x-coordinates
![Page 8: Relations and Functions. Review A relation between two variables x and y is a set of ordered pairs An ordered pair consist of a x and y-coordinate A relation.](https://reader035.fdocuments.us/reader035/viewer/2022070410/56649f1f5503460f94c36dc4/html5/thumbnails/8.jpg)
Example 4
{(– , ), ( , ), ( , ), ( , ), ( , ), (– , )} 1 7 1 0 2 3 0 8 0 5 2 1
•Is this a function?•Hint: Look only at the x-coordinates
NO
![Page 9: Relations and Functions. Review A relation between two variables x and y is a set of ordered pairs An ordered pair consist of a x and y-coordinate A relation.](https://reader035.fdocuments.us/reader035/viewer/2022070410/56649f1f5503460f94c36dc4/html5/thumbnails/9.jpg)
Choice One Choice Two
Example 5
310
–123
2–13
23–20
Which mapping represents a function?
Choice 1
![Page 10: Relations and Functions. Review A relation between two variables x and y is a set of ordered pairs An ordered pair consist of a x and y-coordinate A relation.](https://reader035.fdocuments.us/reader035/viewer/2022070410/56649f1f5503460f94c36dc4/html5/thumbnails/10.jpg)
Example 6Which mapping represents a function?
A. B.
B
![Page 11: Relations and Functions. Review A relation between two variables x and y is a set of ordered pairs An ordered pair consist of a x and y-coordinate A relation.](https://reader035.fdocuments.us/reader035/viewer/2022070410/56649f1f5503460f94c36dc4/html5/thumbnails/11.jpg)
Vertical Line Test•Vertical Line Test: a relation is a function if a vertical line drawn through its graph, passes through only one point.
AKA: “The Pencil Test”Take a pencil and move it from left to right (–x to x); if it crosses more than one point, it is not a function
![Page 12: Relations and Functions. Review A relation between two variables x and y is a set of ordered pairs An ordered pair consist of a x and y-coordinate A relation.](https://reader035.fdocuments.us/reader035/viewer/2022070410/56649f1f5503460f94c36dc4/html5/thumbnails/12.jpg)
Vertical Line Test
Would this graph be a function?YES
![Page 13: Relations and Functions. Review A relation between two variables x and y is a set of ordered pairs An ordered pair consist of a x and y-coordinate A relation.](https://reader035.fdocuments.us/reader035/viewer/2022070410/56649f1f5503460f94c36dc4/html5/thumbnails/13.jpg)
Vertical Line Test
Would this graph be a function?
NO
![Page 14: Relations and Functions. Review A relation between two variables x and y is a set of ordered pairs An ordered pair consist of a x and y-coordinate A relation.](https://reader035.fdocuments.us/reader035/viewer/2022070410/56649f1f5503460f94c36dc4/html5/thumbnails/14.jpg)
Function Notationf(x) means function of x and is read “f of x.”
f(x) = 2x + 1 is written in function notation. The notation f(1) means to replace x with 1 resulting in the function value.
f(1) = 2x + 1 f(1) = 2(1) + 1 f(1) = 3
![Page 15: Relations and Functions. Review A relation between two variables x and y is a set of ordered pairs An ordered pair consist of a x and y-coordinate A relation.](https://reader035.fdocuments.us/reader035/viewer/2022070410/56649f1f5503460f94c36dc4/html5/thumbnails/15.jpg)
Function Notation
Given g(x) = x2 – 3, find g(-2) .
g(-2) = x2 – 3g(-2) = (-2)2 – 3g(-2) = 1
![Page 16: Relations and Functions. Review A relation between two variables x and y is a set of ordered pairs An ordered pair consist of a x and y-coordinate A relation.](https://reader035.fdocuments.us/reader035/viewer/2022070410/56649f1f5503460f94c36dc4/html5/thumbnails/16.jpg)
Function Notation
Given f(x) = , the following.
f(3) = 2x2 – 3xf(3) = 2(3)2 – 3(3)f(3) = 2(9) - 9f(3) = 9
a. f(3)
b. 3f(x)
c. f(3x)
3f(x) = 3(2x2 – 3x)3f(x) = 6x2 – 9x
f(3x) = 2x2 – 3xf(3x) = 2(3x)2 – 3(3x)f(3x) = 2(9x2) – 3(3x)f(3x) = 18x2 – 9x
22 3x x
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For each function, evaluate f(0), f(1.5), f(-4),
f(0) =
f(1.5) =
f(-4) =
3
4
4