Answer the following questions about inverse functions without performing any…
MATHEMATICS · Multi-part · A Level · Mathematics · Algebra and Functions · Algebra and Functions
Problem
Answer the following questions about inverse functions without performing any numerical calculations.
(a) State the defining conditions that two functions f and g must satisfy in order to be inverses of each other.
(b) Explain why verifying only one of the two possible compositions (either g∘f or f∘g) equals the identity is not sufficient to conclude that f and g are inverses.
(c) Give an explicit example of two real-valued functions for which g∘f is the identity on the domain of f but f∘g fails to be the identity on the co-domain of g. Briefly indicate the reason for this failure in terms of domain or co-domain restrictions.