AdvancedProAdvanced Functions
Composite Function
f(g(x))
Overview
A composite function applies g first, then f. The output of g becomes the input of f.
Domain
x where g(x) ∈ domain(f)
Range
Range of f applied to range of g
Symmetry
Depends on f and g
Key Properties
1
f(g(x)) ≠ g(f(x)) in general
2
Domain: x where g(x) is in domain of f
3
Chain rule: (f∘g)'(x) = f'(g(x))·g'(x)
4
Composition is associative but not commutative
Key Insight
Think of f(g(x)) as a pipeline: x → g → g(x) → f → f(g(x)). The order matters!
Visualize Composite Function
Interact with the graph, adjust parameters, and see how the function behaves in real time.