1. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §9.2a Composite Fcns

2. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §9.1 → The NATURAL Base, e  Any QUESTIONS About HomeWork §9.1 → HW-43 9.1 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 3 Bruce Mayer, PE Chabot College Mathematics Composite Functions  In the real world, functions frequently occur in which some quantity depends on a variable that, in turn, depends on another variable.  Functions such as these are called COMPOSITE FUNCTIONS

4. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 4 Bruce Mayer, PE Chabot College Mathematics Composing a Function  Composition with sets A & B by fcns g & f 137137 A 4 10 22 B −128−128 C fg h h(x) = ?

5. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 5 Bruce Mayer, PE Chabot College Mathematics Composing a Function  From The Diagram notice that since f takes the output from g we can combine f and g to get a function h: f (g (x)) = f (3x + 1)  This Yields an eqn for h:

6. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 6 Bruce Mayer, PE Chabot College Mathematics Composing a Function  The function h is the composition of f and g and is denoted f○g (read “the composition of f and g,” “f composed with g,” or “f circle g”).

7. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 7 Bruce Mayer, PE Chabot College Mathematics COMPOSITION OF FUNCTIONS  If f and g are two functions, the composition of function f with function g is written as f○g and is defined by the equation  The function where the domain of f○g consists of those values x in the domain of g for which g(x) is in the domain of f

8. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 8 Bruce Mayer, PE Chabot College Mathematics COMPOSITION OF FUNCTIONS  Graphically the f○g Domain Chain

9. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 9 Bruce Mayer, PE Chabot College Mathematics COMPOSITION OF FUNCTIONS  Conceptually the f○g Operation Chain

10. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 10 Bruce Mayer, PE Chabot College Mathematics Example  Evaluate Composites  Given:  Find Each of the Following  Solution a.

11. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 11 Bruce Mayer, PE Chabot College Mathematics Example  Evaluate Composites  Solution b.  Solution c.  Solution d.

12. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 12 Bruce Mayer, PE Chabot College Mathematics Example  Fcn Composition  Given f(x) = 4x and g(x) = x 2 + 2, find  SOLUTION = f (x 2 + 2) = 4(x 2 + 2) = 4x 2 + 8

13. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 13 Bruce Mayer, PE Chabot College Mathematics Example  Fcn Composition  Given f(x) = 4x and g(x) = x 2 + 2, find  SOLUTION = g(4x) = (4x) 2 + 2 = 16x 2 + 2  This example shows that in general

14. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 14 Bruce Mayer, PE Chabot College Mathematics Example  Fcn Composition  Given:  Find Each Composite Function  Solution a.

15. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 15 Bruce Mayer, PE Chabot College Mathematics Example  Fcn Composition  Given:  Solution b.

16. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 16 Bruce Mayer, PE Chabot College Mathematics Example  Fcn Composition  Given:  Solution c.

17. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 17 Bruce Mayer, PE Chabot College Mathematics Example  Composite Domain  Given:  Solution a.

18. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 18 Bruce Mayer, PE Chabot College Mathematics Example  Composite Domain  Given:  Solution b.  Soln c. Domain: (−∞, 0)U(0, ∞) or {x|x ≠ 0}

19. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 19 Bruce Mayer, PE Chabot College Mathematics Example  Composite Domain  Given:  Soln d. Domain: (−∞, −1)U(−1, ∞) or {x|x ≠ −1}

20. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 20 Bruce Mayer, PE Chabot College Mathematics DEcomposing a Function  Given:  Show that each of the following provides a DEcomposition of H(x)

21. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 21 Bruce Mayer, PE Chabot College Mathematics Decomposing a Function  Solution:

22. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 22 Bruce Mayer, PE Chabot College Mathematics Decomposing a Function  Solution:

23. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 23 Bruce Mayer, PE Chabot College Mathematics Example  Automobile Sales  A car dealer offers an 8% discount off the manufacturer’s suggested retail price (MSRP) of x dollars for any new car on his lot. At the same time, the manufacturer offers a $4000 rebate on the purchase. a.Write a function f(x) that represents the price after the rebate. b.Write a function g(x) that represents the price after the dealer’s discount.

24. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 24 Bruce Mayer, PE Chabot College Mathematics Example  Automobile Sales c.Write the Functions (f○g)(x) & (g○f)(x). What do these Functions Represent? d.Calculate (g○f)(x) − (f○g)(x). Interpret this odd-looking expression  Solution a. The MSRP is x dollars, rebate is $4k, so f(x) = x – 4000 represents the price of the car after the rebate.

25. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 25 Bruce Mayer, PE Chabot College Mathematics Example  Automobile Sales  Solution b. The dealer’s discount is 8% of x, or 0.08x, so: g(x) = x – 0.08x = 0.92x represents the price of the car after the dealer’s discount.  Soln c. This represents the price when the DEALER’S discount is is applied first.

26. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 26 Bruce Mayer, PE Chabot College Mathematics Example  Automobile Sales  Solution c. (cont.) This represents the price when the MANUFACTURER’S rebate is applied first.

27. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 27 Bruce Mayer, PE Chabot College Mathematics Example  Automobile Sales  Solution d. This equation shows that it will cost $320 MORE for any car, regardless of its price, if you apply the rebate first and then the discount second.

28. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 28 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §9.2 Exercise Set 10, 12, 56, 58, 70  Composition of Functions Corresponds to a Production Line

29. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 29 Bruce Mayer, PE Chabot College Mathematics All Done for Today Function Machines & CoDomain

30. BMayer@ChabotCollege.edu MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 30 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –