3. AAT-A IB - HR Date: 3/11/2014 ID Check Obj: SWBAT perform function operations.  Bell Ringer: ACT Prep Probs  HW Requests: pg 375 #42, 43; Worksheet on finding roots of Polynomials #1-13 odds, 2, 4, 6 complete the factoring worksheets; pg 387 #17-22, 47, 48  HW: pg 386 #23-45 odds, 50-51 Read Section 7.8  Announcements: “ There is something in every one of you that waits and listens for the sound of the genuine in yourself. It is the only true guide you will ever have. And if you cannot hear it, you will all of your life spend your days on the ends of strings that somebody else pulls.” ― Howard ThurmanHoward Thurman Maximize Academic Potential Turn UP! MAP

4. COMPOSITION OF FUNCTIONS “SUBSTITUTING ONE FUNCTION INTO ANOTHER”

5. The Composition Function This is read “f composition g” or “f of g(x)” and means to copy the f function down but where ever you see an x, substitute in the g function. FOIL first and then distribute the 2

6. This is read “g composition f” or “g of f(x)” and means to copy the g function down but where ever you see an x, substitute in the f function. You could multiply this out but since it’s to the 3 rd power we won’t

7. This is read “f composition f” or “f of f(x)” and means to copy the f function down but where ever you see an x, substitute in the f function. (So sub the function into itself).

8. The DOMAIN of the Composition Function The domain of f composition g is the set of all numbers x in the domain of g such that g(x) is in the domain of f. The domain of g is x  1 We also have to worry about any “illegals” in this composition function, specifically dividing by 0. This would mean that x  1 so the domain of the composition would be combining the two restrictions.

9. Example – Composition of Functions Method 1: Method 2:

10. Let’s try some

11. Solution

12. Solving with a Graphing Calculator Start with the y= list. Input x 3 for Y 1 and x 2 +7 for Y 2 Now go back to the home screen. Press VARS, YVARS and select 1. You will get the list of functions. Using VARS and YVARS enter the function as Y 2 (Y 1 (2). You should get 71 as a solution.

13. Real Life Application You are shopping in a store that is offering 20% off everything. You also have a coupon for $5 off any item. 1.Write functions for the two situations. Let x = original price. – 20% discount: f(x) = x – 0.20x = 0.8x – Cost with the coupon: g(x) = x - 5

14. You are shopping in a store that is offering 20% off everything. You also have a coupon for $5 off any item. 2. Make a composition of functions: This represents if the clerk does the discount first, then takes $5 off the discounted price.

15. You are shopping in a store that is offering 20% off everything. You also have a coupon for $5 off any item. 3. Now try applying the $5 coupon first, then taking 20% off: How much more will it be if the clerk applies the coupon BEFORE the discount?

16. Composite Function – When you combine two or more functions The composition of function g with function is written as 1 2 1. Evaluate the inner function f(x) first. 2. Then use your answer as the input of the outer function g(x).

17. You are shopping in a store that is offering 20% off everything. You also have a coupon for $5 off any item. 4. Subtract the two functions: Any item will be $1 more if the coupon is applied first. You will save $1 if you take the discount, then use the coupon.

18. 7.6 Function Operations

19. Review: What is a function? A relationship where every domain (x value) has exactly one unique range (y value). Sometimes we talk about a FUNCTION MACHINE, where a rule is applied to each input of x

20. Function Operations

21. Adding and Subtracting Functions When we look at functions we also want to look at their domains (valid x values). In this case, the domain is all real numbers.

22. Multiplying Functions In this case, the domain is all real numbers because there are no values that will make the function invalid.

23. Dividing Functions In this case, the domain is all real numbers EXCEPT -1, because x=-1 would give a zero in the denominator.

24. The sum f + g This just says that to find the sum of two functions, add them together. You should simplify by finding like terms. Combine like terms & put in descending order

25. The difference f - g To find the difference between two functions, subtract the first from the second. CAUTION: Make sure you distribute the – to each term of the second function. You should simplify by combining like terms. Distribute negative

26. The product f g To find the product of two functions, put parenthesis around them and multiply each term from the first function to each term of the second function. FOIL Good idea to put in descending order but not required.

27. The quotient f /g To find the quotient of two functions, put the first one over the second. Nothing more you could do here. (If you can reduce these you should). What is the domain?

28. So the first 4 operations on functions are pretty straight forward. The rules for the domain of functions would apply to these combinations of functions as well. The domain of the sum, difference or product would be the numbers x in the domains of both f and g. For the quotient, you would also need to exclude any numbers x that would make the resulting denominator 0.

29. Let’s Try Some What is the domain?

30. Let’s Try Some What is the domain?

31. Let’s Try Some What is the domain?

32. Let’s Try Some What is the domain?