Let and be bijective functions satisfying .

MATHEMATICS · Multi-part · A Level · Mathematics · Algebra and Functions · Algebra and Functions

Problem

Let f:R→Rf:\mathbb R\to\mathbb R and g:R→Rg:\mathbb R\to\mathbb R be bijective functions satisfying g∘f=f∘gg\circ f=f\circ g. (i) Prove that f−1∘g=g∘f−1f^{-1}\circ g=g\circ f^{-1}. (ii) Hence show that (g∘f)−1=f−1∘g−1=g−1∘f−1(g\circ f)^{-1}=f^{-1}\circ g^{-1}=g^{-1}\circ f^{-1}.

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