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Lesson 26 – Composition of Functions Integrated Math 10 – Mr. Santowski 12/24/2015Integrated...
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Transcript of Lesson 26 – Composition of Functions Integrated Math 10 – Mr. Santowski 12/24/2015Integrated...
![Page 1: Lesson 26 – Composition of Functions Integrated Math 10 – Mr. Santowski 12/24/2015Integrated Math 101.](https://reader036.fdocuments.in/reader036/viewer/2022062423/5697bfee1a28abf838cb9a06/html5/thumbnails/1.jpg)
Lesson 26 – Composition of Functions
Integrated Math 10 – Mr. Santowski
04/21/23Integrated Math 101
![Page 2: Lesson 26 – Composition of Functions Integrated Math 10 – Mr. Santowski 12/24/2015Integrated Math 101.](https://reader036.fdocuments.in/reader036/viewer/2022062423/5697bfee1a28abf838cb9a06/html5/thumbnails/2.jpg)
Fast Five – Warm up Questions
04/21/23Integrated Math 102
Given f(x) = 1 – x2, evaluate: (i) f(3) (ii) f(-1) (iii) f(m)
(iv) f(2a – 1)
Given g(x) = ½x – 4, evaluate: (i) f(3) (ii) f(-1) (iii) f(m)
(iv) f(2a – 1)
Given e(x) = 2x + x + 3, evaluate: (i) f(3) (ii) f(-1) (iii) f(m)
(iv) f(2a – 1)
![Page 3: Lesson 26 – Composition of Functions Integrated Math 10 – Mr. Santowski 12/24/2015Integrated Math 101.](https://reader036.fdocuments.in/reader036/viewer/2022062423/5697bfee1a28abf838cb9a06/html5/thumbnails/3.jpg)
Lesson Objectives
04/21/23Integrated Math 103
Introduce composition of functions using a variety of representations
Define composition of functions and notation associated with function composition
Practice foundational skills with function composition
Use composition of function in some real world examples
![Page 4: Lesson 26 – Composition of Functions Integrated Math 10 – Mr. Santowski 12/24/2015Integrated Math 101.](https://reader036.fdocuments.in/reader036/viewer/2022062423/5697bfee1a28abf838cb9a06/html5/thumbnails/4.jpg)
(A) Composition of Functions – An Example
04/21/23Integrated Math 104
The following example will illustrate one ways of understand the composition of functions
Andrew earns a daily wage of $20/h plus $15/d for travel expenses.
We can demonstrate this with a table of values
Hours Worked Daily Earnings
2
3
4
5
6
7
9
![Page 5: Lesson 26 – Composition of Functions Integrated Math 10 – Mr. Santowski 12/24/2015Integrated Math 101.](https://reader036.fdocuments.in/reader036/viewer/2022062423/5697bfee1a28abf838cb9a06/html5/thumbnails/5.jpg)
(A) Composition of Functions – An Example
04/21/23Integrated Math 105
The following example will illustrate one ways of understand the composition of functions
However, Andrew also pays union fees at 2.5% of his daily earnings, which we can write as the equation Fees = 0.025 x (daily earnings)
We can also demonstrate with a table of values
Hours Worked
Daily Earnings
Union Fees Paid
2
3
4
5
6
7
9
![Page 6: Lesson 26 – Composition of Functions Integrated Math 10 – Mr. Santowski 12/24/2015Integrated Math 101.](https://reader036.fdocuments.in/reader036/viewer/2022062423/5697bfee1a28abf838cb9a06/html5/thumbnails/6.jpg)
(A) Composition of Functions – An Example
04/21/23Integrated Math 106
What we see is that the one function value (daily earnings or E) is being substituted into the second function (Fees = 0.025 x daily earnings) in order to generate the value for the union fees.
We can generate a direct formula for the union fees by substituting the earnings function into the Fees function as follows: Fees = 0.025(20h + 15).
Hence, the Fees function is called a composed function as Fees(daily earnings) = 0.025 x daily earnings
![Page 7: Lesson 26 – Composition of Functions Integrated Math 10 – Mr. Santowski 12/24/2015Integrated Math 101.](https://reader036.fdocuments.in/reader036/viewer/2022062423/5697bfee1a28abf838cb9a06/html5/thumbnails/7.jpg)
(B) Definition of composite functions
04/21/23Integrated Math 107
Suppose f and g are functions such that the range of g is the subset of the domain of f.
Then the composite function can be described by the equation
gf
xgfxgfgf
![Page 8: Lesson 26 – Composition of Functions Integrated Math 10 – Mr. Santowski 12/24/2015Integrated Math 101.](https://reader036.fdocuments.in/reader036/viewer/2022062423/5697bfee1a28abf838cb9a06/html5/thumbnails/8.jpg)
(E) Composition of Functions – Example #3
04/21/23Integrated Math 108
We can define f and g differently, this time as formulas:
f(x) = x² - 3 g(x) = 2x + 7
We will try the following:
(i) f(g(3)) or fog(3) (ii) gof(3) or g(f(3)) (ii) fog(x) and gof(x) (ii) evaluate fog (5) (iii) evaluate gof (9) and g(f(7)) and gog (1)
![Page 9: Lesson 26 – Composition of Functions Integrated Math 10 – Mr. Santowski 12/24/2015Integrated Math 101.](https://reader036.fdocuments.in/reader036/viewer/2022062423/5697bfee1a28abf838cb9a06/html5/thumbnails/9.jpg)
(E) Composition of Functions – Example #2
04/21/23Integrated Math 109
ex 2. We will now define f and g as follows:
f = {(3,2), (5,1), (7,4), (9,3), (11,5)} g = {(1,3), (2,5), (3,7), (4,9), (5,10)}
We will evaluate fog(3) (or f(g(3)) ????? (ii) evaluate fog (1) (iii) evaluate fog (5) and see what happens
why? (iv) evaluate gof (9) and g(f(7) and gog (1)
![Page 10: Lesson 26 – Composition of Functions Integrated Math 10 – Mr. Santowski 12/24/2015Integrated Math 101.](https://reader036.fdocuments.in/reader036/viewer/2022062423/5697bfee1a28abf838cb9a06/html5/thumbnails/10.jpg)
(E) Composition of Functions – Example #2
04/21/23Integrated Math 1010
Here’s an example with mappings:
x
-3
-2
1
5
7
9
g
f
![Page 11: Lesson 26 – Composition of Functions Integrated Math 10 – Mr. Santowski 12/24/2015Integrated Math 101.](https://reader036.fdocuments.in/reader036/viewer/2022062423/5697bfee1a28abf838cb9a06/html5/thumbnails/11.jpg)
(E) Composition of Functions – Example #2
04/21/23Integrated Math 1011
Here’s an example with graphs:
http://teachers.henrico.k12.va.us/math/hcpsalgebra2/Geogebra/composition.html
![Page 12: Lesson 26 – Composition of Functions Integrated Math 10 – Mr. Santowski 12/24/2015Integrated Math 101.](https://reader036.fdocuments.in/reader036/viewer/2022062423/5697bfee1a28abf838cb9a06/html5/thumbnails/12.jpg)
(E) Composition of Functions – Example #2
04/21/23Integrated Math 1012
Links to worksheets:
http://www.mrc.stlmath.com/pdf/m131pdf/compfnc.pdf
http://www.mathworksheetsgo.com/downloads/algebra-2/functions-and-relations/composition-of-functions-worksheet.pdf
http://teachers.henrico.k12.va.us/math/hcpsalgebra2/Documents/8-7/CompositeFunctions8_7.pdf
http://teachers.henrico.k12.va.us/math/hcpsalgebra2/Documents/8-7/8_7_HW.pdf
http://academic.cuesta.edu/mturner/m127/ws_comp.pdf
![Page 13: Lesson 26 – Composition of Functions Integrated Math 10 – Mr. Santowski 12/24/2015Integrated Math 101.](https://reader036.fdocuments.in/reader036/viewer/2022062423/5697bfee1a28abf838cb9a06/html5/thumbnails/13.jpg)
(F) Internet Links
04/21/23Integrated Math 1013
READING: Composition of Functions from PurpleMath
Video Links: http://vimeo.com/12958000 http://www.youtube.com/watch?
v=nZfTvdee240&feature=related http://www.onlinemathlearning.com/
composite-functions-2.html http://www.youtube.com/watch?
v=qxBmISCJSME&feature=related