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Find a codomain B such that ƒ: A → B, ƒ(x) = e²´ is surjective. Show that g: B → A, g(x) = Vln x is the inverse of f. Why is f-(x) # -VIn x?

Find a codomain B such that ƒ: A → B, ƒ(x) = e²´ is surjective. Show that g: B → A, g(x) = Vln x is the inverse of f. Why is f-(x) # -VIn x?

Advanced Engineering Mathematics
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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Questions C and D

(a) Explain why the function f(x) = e=° is not injective (one-to-one) on its natural domain. (b) Find the largest possible domain A, where all elements of A are non-negative and f: A → R, f(x) = e** is injective. (c) Find a codomain B such that f: A→ B, f(x) = e*² is surjective. (d) Show that g: B → A, g(x) = Vln x is the inverse of f. Why is f-'(x) # -VIn x?
Transcribed Image Text:(a) Explain why the function f(x) = e=° is not injective (one-to-one) on its natural domain. (b) Find the largest possible domain A, where all elements of A are non-negative and f: A → R, f(x) = e** is injective. (c) Find a codomain B such that f: A→ B, f(x) = e*² is surjective. (d) Show that g: B → A, g(x) = Vln x is the inverse of f. Why is f-'(x) # -VIn x?
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