Class Opener:

• Given: and find each composition:a) fog(x)b) gof(x)

What is the domain of (fog)(x) given:

Identifying a Composite Function:

• Write the following function as a composition of two functions:

Identifying a Composite Function:

• Write the following function as a composition of two functions:

Identifying a Composition of Two Functions:

• Find two functions f and g such that

Where:

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.

a) Find the composition N(T(t)) and interpret its meaning in context.

b) Find the number of bacteria in the food when t = 2c) Find the time when the bacterial count reaches 2000

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.

Finding the Inverse Function Informally

• Find the inverse function of:

• Prove that the inverse function and the original function will produce the identify function.

Finding the Inverse Function Informally

• Find the inverse function for each:

Verifying the Inverse Function Algebraically

• Show that the functions are invers functions of each other:

Verifying the Inverse Function Algebraically

• Which of the functions is the inverse function of

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.

Testing for one to one functions

Is the function: one to one

Testing one to one functions

• Is the function one to one

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.