2. Function Operations and Composition MM2A5d. Use composition to verify that functions are inverses of each other.

3. Review: functions, domain, & range The domain of a function is the set of all “first coordinates” of the ordered pairs of a relation. The range of a function is the set of all “second coordinates” of the ordered pairs of a relation. A relation is a function if all values of the domain are unique (they do not repeat). A test to see if a relation is a function is the vertical line test. –If it is possible to draw a vertical line and cross the graph of a relation in more than one point, the relation is not a function.

4. Example 1: If f(x) = 3x and g(x) = x – 5 a)Find f(x) + g(x) b) Find f(x) - g(x) c) Find f(x) g(x) What about f(x) ÷ g(x) ? You can perform operations, such as addition, subtraction, multiplication, and division, with functions…

5. f(x) ÷ g(x) = 3x x – 5 Be sure to include in your answer restrictions on the domain (the possible inputs)… The domain is does not include 5 since that would make the denominator 0… Therefore, the domain is: all real numbers except x = 5.

6. Example 2: Find the domain of Think of the values that will make the square root a negative number The domain is all real numbers greater than 1. The graph will confirm this...

7. Your Turn!! Find each function and state its domain:Find each function and state its domain: –f + g –f – g –f ·g –f / g

8. What is function composition?? A composition of functions occurs when you insert one function into another. In effect, the range of the one function becomes the domain of the second. The notation for composition of functions is either:

9. Here’s another way to look at it… Function Machine x Function Machine g f

10. Example 1 Example 1 f(x) = 2x 3 g(x) = x -1 Evaluate f(g(x)): **substitute g(x) into f(x) for the x value!!

11. Example 1 cont. Example 1 cont. f(x) = 2x 3 g(x) = x -1 Evaluate g(f(x)): **substitute f(x) into g(x) for the x value!!

12. Example 2 Example 2 f(x) = 3x 2 g(x) = x 2 + 5 Evaluate f(g(x)) and g(f(x)):

13. Example 3 Example 3 f(x) = x + 2 g(x) = x 3 Evaluate f(g(x)) and g(f(x)):

14. Example 4 Example 4 f(x) = 2x + 1 g(x) = Evaluate f(g(x)) and g(f(x)):

15. Time to Practice!!

16. Homework!! Pg. 114: 1 – 24 ALL Write Problems and Show WORK for credit!!