Composition of functions
The composite g ∘ f: when it exists, reading order, non-commutativity, associativity, the identity, the inverse of a composite, and composites of injections or surjections.
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Composite g ∘ f: what is it?
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The function that sends x to g(f(x)): apply f first, then g to the result. It exists only if the values of f lie in the domain of g.
It reads right to left. With f(x) = x + 1 and g(x) = 2x, g ∘ f gives 2x + 2, whereas f ∘ g gives 2x + 1: composition is not commutative.
With f(x) = 2x + 1 and g(x) = x², what is (g ∘ f)(3)?
- 49
- 19
- 16
- 50
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49
Apply f first: f(3) = 7, then g: 7² = 49. 19 is (f ∘ g)(3) = 2 × 9 + 1, the composite in the opposite order.
In g ∘ f, you first apply …, then …: (g ∘ f)(x) = g(f(x)).
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In g ∘ f, you first apply f, then g: (g ∘ f)(x) = g(f(x)).
The notation reads like the expression g(f(x)): the function closest to x acts first.
Associativity of composition: what is it?
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The property (h ∘ g) ∘ f = h ∘ (g ∘ f): when three functions are chained, where the parentheses go does not change the result.
Both sides send x to h(g(f(x))). So h ∘ g ∘ f can be written without parentheses, but the order of the functions cannot be swapped.