By: Dr. Julia Arnold Composition is a binary operation like addition, subtraction, multiplication...
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Transcript of By: Dr. Julia Arnold Composition is a binary operation like addition, subtraction, multiplication...
![Page 1: By: Dr. Julia Arnold Composition is a binary operation like addition, subtraction, multiplication and division are binary operations. (meaning they operate.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551bd3ee550346c3588b563c/html5/thumbnails/1.jpg)
By: Dr. Julia Arnold
![Page 2: By: Dr. Julia Arnold Composition is a binary operation like addition, subtraction, multiplication and division are binary operations. (meaning they operate.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551bd3ee550346c3588b563c/html5/thumbnails/2.jpg)
Composition is a binary operation like addition , subtraction, multiplication and division are binary operations. (meaning they operate on two elements)
f+g
f-g
fg g
f
The composition symbol is: Thus gf
![Page 3: By: Dr. Julia Arnold Composition is a binary operation like addition, subtraction, multiplication and division are binary operations. (meaning they operate.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551bd3ee550346c3588b563c/html5/thumbnails/3.jpg)
![Page 4: By: Dr. Julia Arnold Composition is a binary operation like addition, subtraction, multiplication and division are binary operations. (meaning they operate.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551bd3ee550346c3588b563c/html5/thumbnails/4.jpg)
The easiest way to describe composition is to say it is like substitution. In fact
))(()( xgfxgf
Read f of g of x which means substitute g(x) for x in the f(x) expression.
![Page 5: By: Dr. Julia Arnold Composition is a binary operation like addition, subtraction, multiplication and division are binary operations. (meaning they operate.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551bd3ee550346c3588b563c/html5/thumbnails/5.jpg)
For example:Suppose f(x)= 2x + 3, and
g(x) = 8 - xThen ))(()( xgfxgf
Means substitute the g function forx in the f function… like this
f(x)= 2x + 3
f(g(x) )= 2 g(x) + 3
![Page 6: By: Dr. Julia Arnold Composition is a binary operation like addition, subtraction, multiplication and division are binary operations. (meaning they operate.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551bd3ee550346c3588b563c/html5/thumbnails/6.jpg)
f(x)= 2x + 3, and g(x) = 8 - x ))(()( xgfxgf
f(x)= 2x + 3
f(g(x) )= 2 g(x) + 3Nowsubstitute what g equals for g(x)
f(8 - x)= 2 (8 - x) + 3
= 16 - 2x + 3= 19 - 2x
So, = 19 - 2x xgf
![Page 7: By: Dr. Julia Arnold Composition is a binary operation like addition, subtraction, multiplication and division are binary operations. (meaning they operate.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551bd3ee550346c3588b563c/html5/thumbnails/7.jpg)
An interesting fact is that
xfgxgf most of the time.
Let’s see if this is thecase for theprevious example.
![Page 8: By: Dr. Julia Arnold Composition is a binary operation like addition, subtraction, multiplication and division are binary operations. (meaning they operate.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551bd3ee550346c3588b563c/html5/thumbnails/8.jpg)
f(x) = 2x + 3, andg(x) = 8 - x
))(( xfgxfg Thus we will substitute f into g.
g(x) = 8 - x
g(f(x) ) = 8 - f(x)
Nowsubstitutewhat f(x) is:
g(2x + 3) = 8 - (2x + 3)
= 8 - 2x - 3= 5 - 2x
![Page 9: By: Dr. Julia Arnold Composition is a binary operation like addition, subtraction, multiplication and division are binary operations. (meaning they operate.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551bd3ee550346c3588b563c/html5/thumbnails/9.jpg)
![Page 10: By: Dr. Julia Arnold Composition is a binary operation like addition, subtraction, multiplication and division are binary operations. (meaning they operate.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551bd3ee550346c3588b563c/html5/thumbnails/10.jpg)
Let 32)( 2 xxxfand
xxg )(
![Page 11: By: Dr. Julia Arnold Composition is a binary operation like addition, subtraction, multiplication and division are binary operations. (meaning they operate.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551bd3ee550346c3588b563c/html5/thumbnails/11.jpg)
32)( 2 xxxf xxg )(
))(()( xgfxgf
Write the f function 32)( 2 xxxfSubstitute g(x) for x 3)(2)())(( 2 xgxgxgf
Replace g(x) with x 32)(2
xxxf
Simplify 32))(( xxxgf
Step 1
Step 2
Step 3
Step 4
![Page 12: By: Dr. Julia Arnold Composition is a binary operation like addition, subtraction, multiplication and division are binary operations. (meaning they operate.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551bd3ee550346c3588b563c/html5/thumbnails/12.jpg)
32)( 2 xxxf xxg )(
Find: xfg When ready click your mouse.
The answer is:
A) 322 xx
B) 32 xxMove your mouse overthe correct answer.
![Page 13: By: Dr. Julia Arnold Composition is a binary operation like addition, subtraction, multiplication and division are binary operations. (meaning they operate.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551bd3ee550346c3588b563c/html5/thumbnails/13.jpg)
xxxf 2)(xx
xg
2
1)(
Find: xgf When ready click your mouse.
The answer is:
B) xxxx
2
2
2
11
A)xxxx
224
11
Move your mouse overthe correct answer.
![Page 14: By: Dr. Julia Arnold Composition is a binary operation like addition, subtraction, multiplication and division are binary operations. (meaning they operate.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551bd3ee550346c3588b563c/html5/thumbnails/14.jpg)
1)( xxf 21
)( x
xg
Find: xfg When ready click your mouse.
The answer is:
B) 2
1
1
x
A) 121
121
x
x
xMove your mouse overthe correct answer.
![Page 15: By: Dr. Julia Arnold Composition is a binary operation like addition, subtraction, multiplication and division are binary operations. (meaning they operate.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551bd3ee550346c3588b563c/html5/thumbnails/15.jpg)
1)( xxf 21
)( x
xg
A) 121
121
x
x
x
Ans. A for the previous example
Was actually xgf
![Page 16: By: Dr. Julia Arnold Composition is a binary operation like addition, subtraction, multiplication and division are binary operations. (meaning they operate.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551bd3ee550346c3588b563c/html5/thumbnails/16.jpg)
Do the practice problems followingThis module in your module section Of Blackboard. Check your work usingThis live math page.