Lesson 26 – Composition of
Functions
Integrated Math 10 – Mr. Santowski
1
Integrated Math 10
3/23/2016
Fast Five – Warm up Questions
Given f(x) = 1 – x2, evaluate:


(i) f(3)
(ii) f(-1)
(iii) f(m)
(iv) f(2a – 1)
(iii) f(m)
(iv) f(2a – 1)
Given g(x) = ½x – 4, evaluate:


(i) f(3)
(ii) f(-1)
Given e(x) = 2x + x + 3, evaluate:


2
(i) f(3)
(ii) f(-1)
(iii) f(m)
Integrated Math 10
(iv) f(2a – 1)
3/23/2016
Lesson Objectives
3

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
Integrated Math 10
3/23/2016
(A) Composition of Functions – An Example
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

4
Hours Worked
Daily Earnings
2
3
4
5
6
7
9
Integrated Math 10
3/23/2016
(A) Composition of Functions – An Example
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)

Hours
Worked
Daily
Earnings
2
3
4
5
6
7
We can also demonstrate
with a table of values

5
9
Integrated Math 10
3/23/2016
Union
Fees Paid
(A) Composition of Functions – An Example

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
6
Integrated Math 10
3/23/2016
(B) Definition of composite functions

Suppose f and g are functions such that the range of g
is the subset of the domain of f.

Then the composite function
described by the equation
f g
can be
f  g   f  g x   f g x 
7
Integrated Math 10
3/23/2016
(E) Composition of Functions – Example #3
 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)
8
Integrated Math 10
3/23/2016
(E) Composition of Functions – Example #2

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)




9
Integrated Math 10
3/23/2016
(E) Composition of Functions – Example #2

Here’s an example with mappings:
g
x
f
-3
-2
1
5
7
9
10
Integrated Math 10
3/23/2016
(E) Composition of Functions – Example #2

Here’s an example with graphs:

http://teachers.henrico.k12.va.us/math/hcpsalgebra2/Geog
ebra/composition.html
11
Integrated Math 10
3/23/2016
(E) Composition of Functions – Example #2

Links to worksheets:

http://www.mrc.stlmath.com/pdf/m131pdf/compfnc.pdf
http://www.mathworksheetsgo.com/downloads/algebra2/functions-and-relations/composition-of-functionsworksheet.pdf
http://teachers.henrico.k12.va.us/math/hcpsalgebra2/Docu
ments/8-7/CompositeFunctions8_7.pdf
http://teachers.henrico.k12.va.us/math/hcpsalgebra2/Docu
ments/8-7/8_7_HW.pdf
http://academic.cuesta.edu/mturner/m127/ws_comp.pdf




12
Integrated Math 10
3/23/2016
(F) Internet Links







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-functions2.html
http://www.youtube.com/watch?v=qxBmISCJSME&feature
=related
13
Integrated Math 10
3/23/2016