Lesson 26 – Composition of Functions Integrated Math 10 – Mr. Santowski 12/24/2015Integrated Math 101

Fast Five – Warm up Questions 12/24/2015Integrated Math 102 Given f(x) = 1 – x 2, 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) = 2 x + x + 3, evaluate: (i) f(3)(ii) f(-1)(iii) f(m)(iv) f(2a – 1)

Lesson Objectives 12/24/2015Integrated 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 12/24/2015Integrated 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 WorkedDaily Earnings

(A) Composition of Functions – An Example 12/24/2015Integrated 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 = x (daily earnings)  We can also demonstrate with a table of values Hours Worked Daily Earnings Union Fees Paid

(A) Composition of Functions – An Example 12/24/2015Integrated Math 106  What we see is that the one function value (daily earnings or E) is being substituted into the second function (Fees = 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) = x daily earnings

(B) Definition of composite functions 12/24/2015Integrated 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

(E) Composition of Functions – Example #3 12/24/2015Integrated 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 12/24/2015Integrated 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 12/24/2015Integrated Math 1010  Here’s an example with mappings: x g f

(E) Composition of Functions – Example #2 12/24/2015Integrated Math 1011  Here’s an example with graphs:  ebra/composition.html ebra/composition.html

(E) Composition of Functions – Example #2 12/24/2015Integrated Math 1012  Links to worksheets:   2/functions-and-relations/composition-of-functions- worksheet.pdf 2/functions-and-relations/composition-of-functions- worksheet.pdf  ments/8-7/CompositeFunctions8_7.pdf ments/8-7/CompositeFunctions8_7.pdf  ments/8-7/8_7_HW.pdf ments/8-7/8_7_HW.pdf 

(F) Internet Links 12/24/2015Integrated Math 1013  READING:  Composition of Functions from PurpleMath Composition of Functions from PurpleMath  Video Links:   =related =related  2.html 2.html  =related =related