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Lesson 26 – Composition of Functions

Lesson 26 – Composition of Functions. Integrated Math 10 – Mr. Santowski. Fast Five – Warm up Questions. 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)

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Lesson 26 – Composition of Functions

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  1. Lesson 26–Composition of Functions Integrated Math 10 – Mr. Santowski Integrated Math 10

  2. Fast Five – Warm up Questions • 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) Integrated Math 10

  3. Lesson Objectives • 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

  4. (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 Integrated Math 10

  5. (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) • We can also demonstrate with a table of values Integrated Math 10

  6. (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 Integrated Math 10

  7. (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 can be described by the equation Integrated Math 10

  8. (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) Integrated Math 10

  9. (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) Integrated Math 10

  10. (E) Composition of Functions – Example #2 • Here’s an example with mappings: Integrated Math 10

  11. (E) Composition of Functions – Example #2 • Here’s an example with graphs: • http://teachers.henrico.k12.va.us/math/hcpsalgebra2/Geogebra/composition.html Integrated Math 10

  12. (E) Composition of Functions – Example #2 • 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 Integrated Math 10

  13. (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-functions-2.html • http://www.youtube.com/watch?v=qxBmISCJSME&feature=related Integrated Math 10

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