1. 7.6 Function Operations

2. 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

3. Function Operations

4. 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.

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

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

7. Let’s Try Some What is the domain?

8. Let’s Try Some What is the domain?

9. Let’s Try Some What is the domain?

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

11. 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).

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

13. Let’s try some

14. Solution

15. 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.

16. 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

17. 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 they clerk does the discount first, then takes $5 off the discounted price.

18. 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?

19. 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.