Lesson 26 – Composition of Functions
Integrated Math 10 – Mr. Santowski
04/21/23Integrated Math 101
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)
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
(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
(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
(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
(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
(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)
(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)
(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
(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
(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
(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