Class Opener:

1. Class Opener: • Given: and find each composition: • fog(x) • gof(x) What is the domain of (fog)(x) given:

2. Identifying a Composite Function: • Write the following function as a composition of two functions:

3. Identifying a Composite Function: • Write the following function as a composition of two functions:

4. Identifying a Composition of Two Functions: • Find two functions f and g such that Where:

5. ORQ Practice: Bacteria Count: The number N of bacteria in a refrigerated food is given by Where T is the temperature of the food. When the food is removed from the refrigeration, the temperature of the food is given by T(t) = 4t + 2, Where t is the time in hours. • Find the composition N(T(t)) and interpret its meaning in context. • Find the number of bacteria in the food when t = 2 • Find the time when the bacterial count reaches 2000

6. Inverse Functions: • An inverse function is a function from Set B to Set A, and is denoted by • The domain of the original function f is equal to the range of , and vice versa. • The composition of f and will result in the identify function.

7. Finding the Inverse Function Informally • Find the inverse function of: • Prove that the inverse function and the original function will produce the identify function.

8. Finding the Inverse Function Informally • Find the inverse function for each:

9. Verifying the Inverse Function Algebraically • Show that the functions are invers functions of each other:

10. Verifying the Inverse Function Algebraically • Which of the functions is the inverse function of

11. One – to – One Functions • A function is one to one if, for a and b in its domain, f(a) = f(b) implies that a = b. • A function f has an inverse function if and only if f is one to one.

12. Testing for one to one functions Is the function: one to one

13. Testing one to one functions • Is the function one to one

14. Horizontal Line Test • Use the horizontal line test on a graph of a function to see if it is one to one. • If it is a function the horizontal line will only hit the function one time.