Algebra and Trigonometry - Exponential and Logarithmic Functions

Algebra and Trigonometry - Chapter 6: Exponential and Logarithmic Functions

Section 6.1: Composite Functions

Objectives

  • Form a Composite Function
  • Find the Domain of a Composite Function

Composite Function Definition

  • Given two functions f and g, the composite function, denoted by (f ∘ g) (read as “f composed with g”), is defined as:
    • (f ∘ g)(x) = f(g(x))
  • Domain of Composite Function:
    • The domain of (f ∘ g) is the set of all numbers x in the domain of g for which g(x) is in the domain of f.

Example 1: Evaluating a Composite Function

  • Given Functions:
    • f(x) and g(x) (specific functions not provided)
  • Find:
    • a) Calculate (f ∘ g)(x)
    • b) Calculate (g ∘ f)(x)
    • c) Evaluate (f ∘ g)(c) (where c is a specific value)
    • d) Evaluate (g ∘ f)(c).

Example 2: Finding a Composite Function and Its Domain

  • Given Functions:
    • f(x) and g(x) (functions not specified)
  • Tasks:
    • a) Find (f ∘ g)(x)
    • b) Find (g ∘ f)(x)
  • Domain of each composite function:
    • The domain of f and g are both the set of all real numbers. Thus, the domain of both (f ∘ g) and (g ∘ f) is the set of all real numbers.

Example 3: Finding the Domain of f Composition g

  • Objective: Find the domain of (f ∘ g)(x).
  • Steps:
    1. Identify the domain of g. Exclude values as necessary, e.g., exclude 3 from the domain of g due to restrictions.
    2. Determine the domain of f where it is not defined, e.g., exclude -4.
    3. Solve an equation to find if other values should be excluded from the domain.
    • For instance, if given:
      f(x) = rac{1}{x - 1}
      Multiply both sides by (x - 1) for clarity.

Example 4: Finding a Composite Function and Its Domain (1 of 5)

  • Given Functions:
    • Specific functions not specified here.
  • Tasks:
    • Find:
    • a) (f ∘ g)(x)
    • b) (g ∘ f)(x)
  • Domain of Functions:
    • f has domain restrictions identified, e.g., exclude 2 from the domain of g due to a division issue.
    • Look into the domain of f and note that if x = 1, it results in division by zero, thus exclude 1 from the domain.
  • Comprehensively analyze the domains to determine restrictions on the composite functions:
    • Define what other numbers to exclude based on further conditions or restrictions through solving related equations.

Example 5: Showing That Two Composite Functions Are Equal (1 of 2)

  • Objective: If h(x) = (f ∘ g)(x), show that for every x in the domain of f and g, h(x) = f(g(x)) holds true.
  • Conclusion:
    • We demonstrate equality across the specified domain.

Example 6: Finding the Components of a Composite Function

  • Objective: Find functions f and g such that (f ∘ g)(x) = H(x) when H raises the expression x to the power 21.
    • Let g(x) = x and f(x) = x^{21}.

Example 7: Finding the Components of a Composite Function

  • Objective: Similar task as previous, but here, H is the reciprocal of the function.
    • Assign appropriate functions f and g such that they satisfy the condition for their composition leading to H.

Notes on Function Composition:

  • Ensure understanding of how to manipulate functions for composition and their respective domains.
  • Always check for undefined behavior in functions (like division by zero or square roots of negative numbers).