1. 6.6 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. Composite Function – When you combine two or more functions
The composition of function g with function is written as 1
1. Evaluate the inner function f(x) first.
2. Then use your answer as the input of the outer function g(x). 2

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

11. Let’s try some

12. Solving with a Graphing Calculator
Start with the y= list.
Input x3 for Y1 and x2+7 for Y2
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 Y2(Y1(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.
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. 2. Make a composition of functions:
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.

15. 3. Now try applying the $5 coupon first, then taking 20% off:
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. 4. Subtract the two functions:
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.