Algebra and Trigonometry: Chapter 6 - Exponential and Logarithmic Functions

Algebra and Trigonometry Twelfth Edition

Chapter 6: Exponential and Logarithmic Functions

Section 6.1: Composite Functions

Objectives
  • Form a Composite Function
  • Find the Domain of a Composite Function
Definition of Composite Function
  • A Composite Function is formed by combining two functions, denoted by \( f(g(x)) \. It reads as "f composed with g".
  • The domain of the composite function \( f(g(x)) \) is the set of all numbers \( x \$ in the domain of \g\ for which \f(g(x)) \$ is in the domain of \f\.

Evaluating Composite Functions

Example 1: Evaluating a Composite Function
  • Suppose we have two functions f and g. To evaluate \( f(g(x)) \$ we perform the following:
    • Step a: Evaluate g(x).
    • Step b: Substitute the result of g(x) into f.
    • Step c: Calculate the final value to find f(g(x)).

Finding Composite Functions and Their Domains

Example 2: Finding a Composite Function and Its Domain
  • Given two functions, we need to:
    • Find \f(g(x)) \$.
    • Determine the domain of the composite function.
  • If both functions f and g have a domain of all real numbers, then the domain of \f(g(x)) \$ is also all real numbers.
Example 3: Finding the Domain of f Composition g
  • Given a composition f(g(x)), the following considerations are important:
    1. Identify restrictions from g(x), for example, if g(x) has a specific value that must be excluded from the domain.
    2. Identify restrictions from f(x).
    3. Create a composite expression and solve it to find additional exclusions.
  • Suppose g has a restriction where 3 is excluded:
    • Let the secondary exclusion come from the equation, such as g equating to a certain value causing another restriction.

Specific Examples of Finding Domains

Example 4: Finding a Composite Function and Its Domain
  • Consider two functions:
    • Given: Domain of f is restricted.
    • First, determine exclusions from g, for example excluding value 2.
    • Check for additional exclusions in f based on its definition, like division by zero case (x cannot equal 1).
  • Specify the domain of the composite function and note necessary exclusions.

Showing Equality of Composite Functions

Example 5: Showing That Two Composite Functions Are Equal
  • To show that two composite functions are equal, consider: if \f(g(x)) = h(x)\, establish their equality by evaluating both sides to confirm identity across the entire domain.

Finding Components of Composite Functions

Example 6: Finding the Components of a Composite Function
  • Given a function \H\(x)\:
    • Suppose \H\ left(x\right) is defined as raising an expression to the power of 21.
    • Define a function f to correspond with part of H, such as (f(x) = x) and let g(x) = the expression being raised to the power 21.
    • Therefore, \H(f(x)) = f(x)^{21}.
Example 7: Finding Another Set of Components
  • Find functions f and g such that \H\(x) = f(g(x))\:
    • Let \H\(x) be defined as the reciprocal of another expression.
    • Define f(x) and g(x) accordingly to satisfy the composition requirement.

Conclusion

  • Composite functions represent a powerful concept in algebra and trigonometry, where understanding how to evaluate them and find their domain is crucial for higher-level mathematics. The notation and execution of finding composite functions and their properties require careful attention to detail, particularly regarding domain restrictions and exclusions.