Notes on Function Composition: Examples and DIY Problem (copy)

Composition of Functions

The operation of applying one function to the results of another, denoted as (f ∘ g)(x) = f(g(x)).

Domain of Composition

The domain of the composite function f ∘ g includes values of x in the domain of g such that g(x) is also in the domain of f.

How to compute (f ∘ g)(x)

1. Identify functions f and g. 2. Compute g(x). 3. Substitute g(x) into f. 4. Simplify the expression.

What is the general formula for the domain of composed functions?

Dom(f ∘ g) = { x ∈ Dom(g) | g(x) ∈ Dom(f) }.

Example of composing functions: f(x) and g(x)

If f(x) = 1/(x+1) and g(x) = 5/(2x), then (f ∘ g)(x) = 2x/(5 + 2x).

Domain of f(x) = 1/(x+1)

Dom(f) = (-∞, -1) ∪ (-1, ∞).

Domain of g(x) = 5/(2x)

Dom(g) = (-∞, 0) ∪ (0, ∞).

What are the simple functions f and g in Example 2?

f(x) = |x|, g(x) = √(2x-1).

When is g(x) = √(2x-1) defined?

2x - 1 ≥ 0, which means x must be ≥ 1/2.

How does the composition h(h(x)) simplify?

h(h(x)) = |x| when h(x) = √(4 - x²).

Domain for h(x) = √(4 - x²)

h is defined for -2 ≤ x ≤ 2.

What is the final result of the composition (h ∘ h)(x)?

(h ∘ h)(x) = |x| for x in [-2, 2].

Key takeaway about composing functions

Always check that the output of g(x) lies within the domain of f before concluding the domain of the composition.