BackFunction Composition and Decomposition: Domains and Formulas
Study Guide - Smart Notes
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Function Composition
Definition and Notation
Function composition is a fundamental concept in precalculus, where two functions are combined such that the output of one function becomes the input of another. The composition of functions f and g is denoted as (f ˆ g)(x) = f(g(x)).
Inner function: The function applied first (g(x)).
Outer function: The function applied to the result of the inner function (f(...)).
Evaluating Compositions
To evaluate a composition, substitute the output of the inner function into the outer function.
Example: If and , then .
Example: .
Domain of Composed Functions
The domain of a composition consists of all in the domain of such that is in the domain of .
Find the domain of the inner function .
Ensure that produces values within the domain of .
Example: If and , then is defined for , and is undefined when (i.e., ). So, the domain is .
Finding and Simplifying Formulas for Compositions
Step-by-Step Process
Write the formula for the inner function .
Substitute into the outer function .
Simplify the resulting expression.
Example: ,
Example: ,
Evaluating Compositions at Specific Values
Substitute the given value into the composed function.
Example:
Example:
Domain Analysis for Compositions
Restrictions from Both Functions
Check the domain of the inner function.
Check for values that make the outer function undefined when composed.
Example: , is defined for , is undefined when (). Domain:
Special Cases
If the inner function's output never reaches the restricted value of the outer function, the domain is just the domain of the inner function.
Example: , Since for all real , the domain is .
Decomposition of Functions
Breaking Down Functions
Decomposition involves expressing a function as the composition of two simpler functions, .
Identify an inner function and an outer function such that reconstructs the original function.
Example: ,
Example: ,
Example: ,
Example: ,
Summary Table: Domains of Composed Functions
Composition | Formula | Domain |
|---|---|---|
Key Points and Applications
Function composition is used to build complex functions from simpler ones.
Domain analysis is crucial to ensure the composition is defined.
Decomposition helps in understanding the structure of functions and is useful in calculus and modeling.
Always check for restrictions from both the inner and outer functions when composing.
Additional info:
All examples and formulas are based on standard precalculus function composition and domain analysis.
Some explanations have been expanded for clarity and completeness.
