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Define the function f:R → R by f(x) = x³ + 1, Vx € R. a) Prove that f is onto? b) Prove that f is one-to-one? c) Explain why f'(x) exists? What is the rule (define) for the inverse function?

Define the function f:R → R by f(x) = x³ + 1, Vx € R. a) Prove that f is onto? b) Prove that f is one-to-one? c) Explain why f'(x) exists? What is the rule (define) for the inverse function?

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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Define the function \( f: \mathbb{R} \to \mathbb{R} \) by \( f(x) = x^3 + 1, \forall x \in \mathbb{R} \). a) Prove that \( f \) is onto? b) Prove that \( f \) is one-to-one? c) Explain why \( f^{-1}(x) \) exists? What is the rule (define) for the inverse function?
Transcribed Image Text:Define the function \( f: \mathbb{R} \to \mathbb{R} \) by \( f(x) = x^3 + 1, \forall x \in \mathbb{R} \). a) Prove that \( f \) is onto? b) Prove that \( f \) is one-to-one? c) Explain why \( f^{-1}(x) \) exists? What is the rule (define) for the inverse function?
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