2. [Falkner Section 11 Exercise 17 - modified] Let f: [1, ∞) → R xx-1. (a) Show that Rng(ƒ) ≤ [0, ∞). That is, ƒ(x) = [0, ∞) for every x € [1,∞). (b) Prove that Rng(ƒ) = [0, ∞). [HINT: In light of part (a), you need only prove the other inclusion, [0, ∞) ≤ Rng(f). That is, for each y = [0, ∞), you must find some x € Dom(f) = [1, ∞) such that f(x) = y.] (c) Prove that f is an injection.
2. [Falkner Section 11 Exercise 17 - modified] Let f: [1, ∞) → R xx-1. (a) Show that Rng(ƒ) ≤ [0, ∞). That is, ƒ(x) = [0, ∞) for every x € [1,∞). (b) Prove that Rng(ƒ) = [0, ∞). [HINT: In light of part (a), you need only prove the other inclusion, [0, ∞) ≤ Rng(f). That is, for each y = [0, ∞), you must find some x € Dom(f) = [1, ∞) such that f(x) = y.] (c) Prove that f is an injection.
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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![2. [Falkner Section 11 Exercise 17 – modified] Let
f: [1, ∞) → R
xx-1.
(a) Show that Rng(ƒ) ≤ [0, ∞). That is, f(x) = [0, ∞) for every x € [1, ∞).
(b) Prove that Rng(ƒ) = [0, ∞). [HINT: In light of part (a), you need only prove the
other inclusion, [0, ∞) ≤ Rng(f). That is, for each y = [0, ∞), you must find
some x € Dom(ƒ) = [1, ∞) such that f(x) = y.]
(c) Prove that f is an injection.
(d) Conclude that ƒ is a bijection from [1, ∞) to [0, ∞), and give a formula for the
inverse function f-¹: [0, ∞) → [1, ∞).
(e) Sketch the graph of f.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd7626d59-6405-4948-a625-19aca32c9eb7%2F31c00ac7-8bf0-479c-9ebd-124f39db17b1%2Fnk8fyvl_processed.png&w=3840&q=75)
Transcribed Image Text:2. [Falkner Section 11 Exercise 17 – modified] Let
f: [1, ∞) → R
xx-1.
(a) Show that Rng(ƒ) ≤ [0, ∞). That is, f(x) = [0, ∞) for every x € [1, ∞).
(b) Prove that Rng(ƒ) = [0, ∞). [HINT: In light of part (a), you need only prove the
other inclusion, [0, ∞) ≤ Rng(f). That is, for each y = [0, ∞), you must find
some x € Dom(ƒ) = [1, ∞) such that f(x) = y.]
(c) Prove that f is an injection.
(d) Conclude that ƒ is a bijection from [1, ∞) to [0, ∞), and give a formula for the
inverse function f-¹: [0, ∞) → [1, ∞).
(e) Sketch the graph of f.
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