Composite Functions Overview

Study Notes

A composite function is formed when one function is applied to the result of another function.
Denoted as (f ∘ g)(x) = f(g(x)), where:
- f is the outer function.
- g is the inner function.
The domain of the composite function is determined by the domain of g and the range of g must be within the domain of f.

Modeling Real-World Scenarios: Used in physics, engineering, and economics to combine different processes.
Complex Functions: Helps simplify the analysis of complex functions by breaking them down into simpler parts.
Data Transformation: Commonly used in statistics and data science to apply multiple transformations sequentially.

Associativity: (f ∘ g) ∘ h = f ∘ (g ∘ h)
Identity: f ∘ id = f and id ∘ f = f, where id is the identity function.
Not Commutative: f ∘ g ≠ g ∘ f in general; order matters.
Continuous and Differentiable: If f and g are continuous, then (f ∘ g) is also continuous.

Can consist of more than two functions: (f ∘ g ∘ h)(x) = f(g(h(x))).
Requires careful tracking of the domain and range for each function involved.
Can be visualized as a chain where each function's output becomes the input for the next.

The inverse of a function f, denoted f⁻¹, satisfies f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.
For a composite function to have an inverse, both functions must be bijective (one-to-one and onto).
If f and g are inverses, then (f ∘ g)(x) = x and (g ∘ f)(x) = x for all x in their respective domains.

A composite function is created when one function is applied to the outcome of another function.
Denotes as (f ∘ g)(x) = f(g(x)):
- f represents the outer function.
- g represents the inner function.
The composite function's domain relies on the domain of g, with the range of g needing to fit within the domain of f.

Employed in fields such as physics, engineering, and economics for modeling real-world scenarios by combining different operations.
Useful for simplifying complex functions, allowing for easier analysis by breaking them into simpler components.
Utilized in statistics and data science for sequential application of multiple transformations on data.

Associativity: The composition of functions is associative, which means (f ∘ g) ∘ h equals f ∘ (g ∘ h).
Identity: Composing any function f with the identity function id yields the original function: f ∘ id = f and id ∘ f = f.
Non-Commutative: Composition is generally not commutative; order significantly impacts results (f ∘ g does not equal g ∘ f).
Continuity and Differentiability: If both f and g are continuous functions, the composite function (f ∘ g) remains continuous.

More than two functions can be composed, expressed as (f ∘ g ∘ h)(x) = f(g(h(x))).
Requires careful attention to the domain and range of all involved functions to ensure correctness.
Visualized as a chain process where each function's output serves as the input for the following function.

A function's inverse, f⁻¹, satisfies the properties: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.
To possess an inverse, both functions in a composite must be bijective (one-to-one and onto).
If f and g are inverses, their composition returns the original input: (f ∘ g)(x) = x and (g ∘ f)(x) = x for all x in their specific domains.