Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

1. Chabot Mathematics §9.2aComposite Fcns Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. MTH 55 9.1 Review § • Any QUESTIONS About • §9.1 → The NATURAL Base, e • Any QUESTIONS About HomeWork • §9.1 → HW-43

3. 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. Composing a Function • Composition with sets A & B by fcns g & f g f A B C 1 3 7 4 10 22 −1 2 8 h(x) = ? h

5. 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. 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. 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 off○gconsists of those values x in the domain of g for which g(x) is in the domain of f

8. COMPOSITION OF FUNCTIONS • Graphically thef○gDomain Chain

9. COMPOSITION OF FUNCTIONS • Conceptually the f○g Operation Chain

10. Example  Evaluate Composites • Given: • Find Each of the Following • Solution a.

11. Example  Evaluate Composites • Solution b. • Solution c. • Solution d.

12. Example  Fcn Composition • Given f(x) = 4x and g(x) = x2 + 2, find • SOLUTION = f (x2 + 2) = 4(x2 + 2) = 4x2 + 8

13. Example  Fcn Composition • Given f(x) = 4x and g(x) = x2 + 2, find • SOLUTION = g(4x) = (4x)2 + 2 = 16x2 + 2 • This example shows that in general

14. Example  Fcn Composition • Given: • Find Each Composite Function • Solution a.

15. Example  Fcn Composition • Given: • Solution b.

16. Example  Fcn Composition • Given: • Solution c.

17. Example  Composite Domain • Given: • Solution a.

18. Example  Composite Domain • Given: • Solution b. • Soln c. • Domain: (−∞, 0)U(0, ∞) or {x|x ≠ 0}

19. Example  Composite Domain • Given: • Soln d. • Domain: (−∞, −1)U(−1, ∞) or {x|x ≠ −1}

20. DEcomposing a Function • Given: • Show that each of the following provides a DEcomposition of H(x)

21. Decomposing a Function • Solution:

22. Decomposing a Function • Solution:

23. 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. • Write a function f(x) that represents the price after the rebate. • Write a function g(x) that represents the price after the dealer’s discount.

24. Example  Automobile Sales • Write the Functions (f○g)(x) & (g○f)(x). What do these Functions Represent? • 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. 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. Example  Automobile Sales • Solution c. (cont.) • This represents the price when the MANUFACTURER’S rebate is applied first.

27. 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. WhiteBoard Work • Problems From §9.2 Exercise Set • 10, 12, 56, 58, 70 • Composition of Functions Corresponds to a Production Line

29. All Done for Today FunctionMachines& CoDomain

30. Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –