5.6.15. Let f: A → B be an arbitrary function. (a) Prove that if f is a bijection (and hence invertible), then f-'(f(x)) = x for all æ E A, and f(f-'(x)) = x for all x E B. (b) Conversely, show that if there is a function g: B → A, satisfying g(f(x)) = x for all x € A, and f(g(x)) = x for all x E B, then f is a bijection, and f = g.
5.6.15. Let f: A → B be an arbitrary function. (a) Prove that if f is a bijection (and hence invertible), then f-'(f(x)) = x for all æ E A, and f(f-'(x)) = x for all x E B. (b) Conversely, show that if there is a function g: B → A, satisfying g(f(x)) = x for all x € A, and f(g(x)) = x for all x E B, then f is a bijection, and f = g.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:5.6.15. Let f: A → B be an arbitrary function.
(a) Prove that if f is a bijection (and hence invertible), then f-'(f(x)) = x for all x E A, and f(ƒ¯'(x))=
x for all x E B.
(b) Conversely, show that if there is a function g: B → A, satisfying g(f(x))
= x for all x € A, and
f(g(x)) = x for all x E B, then f is a bijection, and f-1 = g.
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