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