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Define f : [0, 0) →R by f(x) = x. Find a function g: R → R such that gof is one-to-one, but g is not one-to-one.

Define f : [0, 0) →R by f(x) = x. Find a function g: R → R such that gof is one-to-one, but g is not one-to-one.

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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Please do part g. Please show step by step and explain

(b) Suppose f: A → B and g: B → C. Show that if gof is one-to-one, then f is one-to-one. (c) Suppose f: A → B and g: B → C. Show that if g of is onto, then g is onto. (d) Give an example of functions f: A → B and g: B → C, such that gof is onto, but f is not onto. (e) Suppose f: A → B and g: B → C. Show that if g of is onto, and g is one-to-one, then ƒ is onto. (f) Suppose f: A → B and g: B → C. Show that if f is onto and gof is 1-1, then g is 1-1. (g) Define f: [0, 0) →R by f(x) = x. Find a function g: R → R such that gof is one-to-one, but g is not one-to-one. (h) Suppose f and g are functions from A to A. If f(a) = a for every a E A, then what are f og and go f?
Transcribed Image Text:(b) Suppose f: A → B and g: B → C. Show that if gof is one-to-one, then f is one-to-one. (c) Suppose f: A → B and g: B → C. Show that if g of is onto, then g is onto. (d) Give an example of functions f: A → B and g: B → C, such that gof is onto, but f is not onto. (e) Suppose f: A → B and g: B → C. Show that if g of is onto, and g is one-to-one, then ƒ is onto. (f) Suppose f: A → B and g: B → C. Show that if f is onto and gof is 1-1, then g is 1-1. (g) Define f: [0, 0) →R by f(x) = x. Find a function g: R → R such that gof is one-to-one, but g is not one-to-one. (h) Suppose f and g are functions from A to A. If f(a) = a for every a E A, then what are f og and go f?
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