# Concept derivation guides

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Concept guide



## [Function composition: the permitted inputs of a restricted composition, evaluation order and non-commutativity](/en/derivations/derivation-mko-composition-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-composition)

Let f:ℝ→ℝ, f(x)=x−3, and g:[0,∞)→ℝ, g(u)=√u. (1) Find the permitted inputs D={x∈dom(f) ∣ f(x)∈dom(g)} of g∘f and prove D=[3,∞); (2) evaluate (g∘f)(7) and (g∘f)(4) by 'f first, then g', and explain why (g∘f)(1) is undefined; (3) show that f∘g is a total composition on [0,∞), and conclude g∘f≠f∘g from (f∘g)(4)=−1≠1=(g∘f)(4); (4) show that composition is not multiplication of function values.

Function composition

derivation-mko-composition-01





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