Topic 2 · IB Maths AA
Composite functions
Pick f and g, then drag an x-value through two function machines to see how f∘g and g∘f differ in formula, graph, domain and range.
(f∘g)(x) = f(g(x)) — apply g first, then f.
Domain of f∘g = {x ∈ dom g : g(x) ∈ dom f}
Range of f∘g = f(range of g ∩ dom f)
Domain:
Range:
Domain:
Range:
Try this
- With f(x) = √x and g(x) = x + 2, compare the two domains, then drag x to −1.5 and see which machine breaks down and at which stage.
- Set f(x) = ln x and g(x) = eˣ: both graphs lie on y = x, but check whether f∘g and g∘f have the same domain and range.
- Set f(x) = 1/x and g(x) = 2x − 1: find the one x-value excluded from the domain of f∘g, and the one y-value missing from the range of g∘f.
Exam tip: to state the domain of f∘g, start from the domain of g and remove any x for which g(x) lands outside the domain of f — never read it only from a simplified formula such as ln(eˣ) = x.