Functions and Their Composition

Definition of Function Composition

Function composition involves applying one function to the results of another. If f and g are functions, the composition (f ˆ g)(x) means applying g to x first, then applying f to the result.

• Notation:

• Order matters: in general.

Example 1: Computing Compositions

• Let and .

• Compute :

  • First, find :

  • Then,

• Compute :

  • First, find :

  • Then,

Decomposing Functions into Simpler Functions

Many complex functions can be written as the composition of two or more simpler functions. This is useful for applying the chain rule in differentiation.

• Example 2: Write each function as a composite of two simpler functions:

  • • Let ,

    • Then

  • • Let ,

    • Then

  • • Let ,

    • Then

  • • Let ,

    • Then

The Chain Rule

Statement of the Chain Rule

The chain rule is a fundamental rule for finding the derivative of a composite function. If , then the derivative of with respect to is:

• Formula:

• This means: Differentiate the outer function (with respect to the inner function), then multiply by the derivative of the inner function.

Examples: Applying the Chain Rule

• Example 3: Compute the derivative of the functions in Example 2.

  • • Let , so

  • • Let ,

  • • Let ,

  • • Let ,

Further Examples

• Example 4: Compute the derivative of the following functions:

  • • Let ,

  • • Product rule and chain rule needed:

  • • Let ,

  • • Quotient rule and chain rule needed:

Applications: Rate of Change in Exponential Models

Example 5: Temperature Change in a Yeast Culture

Suppose a yeast culture at room temperature (68°F) is placed in a refrigerator at 38°F. The temperature after hours is given by:

• Find the rate of change of temperature at and hours.

• Differentiate with respect to :

• At :

  • (rounded)

• At :

  • (rounded)

• Interpretation: The negative sign indicates the temperature is decreasing. The rate slows as time increases.

Summary Table: Chain Rule Applications

Additional info: The above notes expand on the chain rule, function composition, and their applications, including examples relevant to business calculus such as exponential decay in temperature models.