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We have seen that the composition of two bijective functions is again bijective. Moreover, bijective functions are always invertible. The following proposition establishes the link between the composition of functions and invertible functions.

Proposition: The Inverse Of a Composition

Let $f:A\mapsto B$ and $g:B\mapsto C$ be bijective functions. Then the composition $g\circ f$ is invertible and its inverse can be calculated as $(g\circ f)^{-1}=f^{-1}\circ g^{-1}.$

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Bibliography

1. Wille, D; Holz, M: "Repetitorium der Linearen Algebra", Binomi Verlag, 1994