college algebra sponsored in part by acee and nsf
DESCRIPTION
COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF. By Vicki Norwich and Jacci White. Activity 1. FUNCTIONS. FUNCTIONS. A function is a rule that assigns a single output to each input. Definition: A relation that assigns to each member of its domain exactly one member, its range. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/1.jpg)
COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF
By Vicki Norwich andJacci White
![Page 2: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/2.jpg)
2
Activity 1
FUNCTIONS
![Page 3: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/3.jpg)
3Functions
FUNCTIONS
• A function is a rule that assigns a single output to each input.
• Definition: A relation that assigns to each member of its domain exactly one member, its range.
• Vertical line test: If it is possible for a vertical line to intersect a graph more than once, the graph is not the graph of a function.
![Page 4: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/4.jpg)
4Functions
This example is a function because no matter where you draw a vertical line it crosses the graph no more than 1 time.
![Page 5: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/5.jpg)
5Functions
This example is not a function because if you draw a vertical line anywhere near the middle of the graph, it will cross more than one time.
![Page 6: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/6.jpg)
6Functions
Practice: Is this the graph of a function?
![Page 7: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/7.jpg)
7Functions
This example is a function because no matter where you draw a vertical line it crosses the graph no more than 1 time.
![Page 8: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/8.jpg)
8Functions
Practice: Is this the graph of a function?
![Page 9: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/9.jpg)
9Functions
This example is not a function because if you draw a vertical line anywhere, it will cross the graph more than one time. near the middle of
![Page 10: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/10.jpg)
10Functions
Practice: Is this the graph of a function?
![Page 11: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/11.jpg)
11Functions
This example is a function because no matter where you draw a vertical line it crosses the graph no more than 1 time.
![Page 12: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/12.jpg)
12Functions
Practice: Is this the graph of a function?
![Page 13: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/13.jpg)
13Functions
This example is a function because no matter where you draw a vertical line it crosses the graph no more than 1 time.
![Page 14: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/14.jpg)
14Functions
Practice: Is this the graph of a function?
![Page 15: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/15.jpg)
15Functions
This example is not a function because if you draw a vertical line anywhere on the right side of the graph, it will cross more than one time.
![Page 16: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/16.jpg)
16Functions
DOMAIN
•The Denominator cannot equal zero. So the domain is made up of all real numbers that will not make the denominator equal to zero
![Page 17: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/17.jpg)
17Functions
Examples
• The domain of this function is the set of all real numbers not equal to 3.
3
7)(
x
xxf
![Page 18: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/18.jpg)
18Functions
What is the domain of f(x)?
2
1)(
2
x
xxf
![Page 19: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/19.jpg)
19Functions
Answer
• The domain of the prior function is the set of all real numbers not equal to 2.
![Page 20: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/20.jpg)
20Functions
Another way to write the answer is:
}2:{ xx
![Page 21: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/21.jpg)
21Functions
Another example of a function and the domain.
}1,1:{
)1)(1(
)2(
1
2)(
2
2
3
xx
domain
xx
xx
x
xxxf
function
![Page 22: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/22.jpg)
22Functions
Practice: What is the domain of the following function?
)4)(3(
12)(
2
xxx
xxxf
![Page 23: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/23.jpg)
23Functions
Your answers should be:
• The set of all real numbers not equal to 0,4, and -3.
}3,4,0:{ xx
![Page 24: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/24.jpg)
24Functions
Domain
• An even index must have a radicand greater than or equal to zero. In other words, you cannot take an even root of a negative number.
![Page 25: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/25.jpg)
25Functions
Examples
• The following function is a square root function. Because square root is even, the part of the function under the square root sign must be greater than or equal to zero.
4)( xxf
![Page 26: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/26.jpg)
26
Solution
• To find the domain of a function that has an even index, set the part under the radical greater than or equal to 0 and solve for x.
• Therefore, the answer is all real numbers x>4
4
04
x
so
x
![Page 27: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/27.jpg)
27Functions
Practice: What is the domain of the following function?
32)( xxg
![Page 28: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/28.jpg)
28Functions
Your answers should be:
}2
3:{
2
3
032
xxanswer
x
so
x
![Page 29: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/29.jpg)
29Functions
All rules for domains must be used whenever they apply.
• What is the domain of f(x)=x-2?
• What is the domain of g(x)=4x+7
• What is the domain of s(t)=(x+5)
(x-2)?
![Page 30: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/30.jpg)
30
Solution: All real numbers
The solution is all real numbers for each of the prior three examples because there are no denominators and no radicals.
![Page 31: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/31.jpg)
31Functions
Practice: What are the domains of the following functions?
1.
2.
3.
9
4)(
2
t
tts
92)( 3 xxxf
xxg 32)(
![Page 32: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/32.jpg)
32Functions
Your answers should be:
1.
2. All real numbers
3. }0:{
}3:{
xx
tt
![Page 33: COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF](https://reader036.vdocuments.us/reader036/viewer/2022062500/56815013550346895dbdfa9c/html5/thumbnails/33.jpg)
33
Reason for the last three answers.
• In the first problem you have to factor the denominator to see when it will equal zero.
• The second function has no fractions (denominator) and no radicals so the answer is all real numbers.
• In the last problem you must set the 3x that is under the radical sign greater than or equal to zero and solve for x by dividing by 3 on both sides.