Composition of Functions Notes (copy)

Composition of functions

A method of combining functions by using the output of one as the input for another.

Notation for composition of functions

Denoted as (f ∘ g)(x) = f(g(x)), meaning apply g to x, then apply f to that result.

Inner function

The function that is applied first in a composition, denoted as g in (f ∘ g)(x).

Outer function

The function that is applied second in a composition, denoted as f in (f ∘ g)(x).

Key observation about function composition

The order of composition matters; (f ∘ g)(x) generally differs from (g ∘ f)(x).

Domain awareness in composition

The range of the inner function must lie within the domain of the outer function for composition to be defined.

Example of a radical function in composition

f(x) = √(x + 1) represents a square root function.

Example of a linear function in composition

g(x) = 3x - 1 represents a linear equation.

Data processing pipelines

Chaining functions together to process data in sequence, where the output of one function serves as the input to another.

Chain rule in calculus

A method to differentiate composed functions, demonstrating the connection between composition and differentiation.