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Consider the function f : R → [-1,+∞) defined by f (x) = x² + 2x. (a) Show that the image (range) of f is [–1, +∞). What is f-'({0})? f-"({-4})? ƒ-'({-1})? Is the function f injective, surjective, and/or bijective? ' Now consider the function g : R → [-1,+∞) defined by g(x) = f(e"). (b) (c) (d) Is g surjective?

Consider the function f : R → [-1,+∞) defined by f (x) = x² + 2x. (a) Show that the image (range) of f is [–1, +∞). What is f-'({0})? f-"({-4})? ƒ-'({-1})? Is the function f injective, surjective, and/or bijective? ' Now consider the function g : R → [-1,+∞) defined by g(x) = f(e"). (b) (c) (d) Is g surjective?

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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Consider the function f : R → [-1,+∞) defined by f (x) = x² + 2x. %3D Show that the image (range) of ƒ is [–1,+∞). What is f-'({0})? ƒ-'({-4})? ƒ-'({-1})? Is the function f injective, surjective, and/or bijective? (a) (b) (c) Now consider the function g : R → [-1,+∞) defined by g(x) (d) Is g surjective? f(e*).
Transcribed Image Text:Consider the function f : R → [-1,+∞) defined by f (x) = x² + 2x. %3D Show that the image (range) of ƒ is [–1,+∞). What is f-'({0})? ƒ-'({-4})? ƒ-'({-1})? Is the function f injective, surjective, and/or bijective? (a) (b) (c) Now consider the function g : R → [-1,+∞) defined by g(x) (d) Is g surjective? f(e*).
Let n be an odd natural number and a1, a2,..., an E {1,2,..., n} all distinct (that is a; † a; for i # j). Prove, using the pigeonhole principle, that x = (1 – a1) · (2 – a2) · ....· (n – an) is even. Remark: Pay attention that you may not choose the numbers a1, a2, . .., an, so showing that this proposition holds for one particular choice of numbers is not sufficient.
Transcribed Image Text:Let n be an odd natural number and a1, a2,..., an E {1,2,..., n} all distinct (that is a; † a; for i # j). Prove, using the pigeonhole principle, that x = (1 – a1) · (2 – a2) · ....· (n – an) is even. Remark: Pay attention that you may not choose the numbers a1, a2, . .., an, so showing that this proposition holds for one particular choice of numbers is not sufficient.
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