Prove or solve the following statements: a. Prove the function f :R - {1} ----> R - {1} defined by f(x) = [(x+1)/(x-1)]^3 is bijective. b. This question concerns functions f : {A,B,C,D,E} ----> {1,2,3,4,5,6,7}. How many such functions are there? How many of these functions are injective? How many are surjective? How many are bijective? c. Suppose A = {a,b,c}. Let f :A ----> A be the function f = {(a,c),(b,c),(c,c)}, and let g: A ----> A be the function g = {(a,a),(b,b),(c,a)}. Find g o f and f o g. d. Consider the functions f,g: Z x Z ----> Z x Z defined as f(m,n) = (3m - 4n,2m + n)and g(m,n) = (5m + n,m). Find the formulas for g o f and f o g.
Prove or solve the following statements: a. Prove the function f :R - {1} ----> R - {1} defined by f(x) = [(x+1)/(x-1)]^3 is bijective. b. This question concerns functions f : {A,B,C,D,E} ----> {1,2,3,4,5,6,7}. How many such functions are there? How many of these functions are injective? How many are surjective? How many are bijective? c. Suppose A = {a,b,c}. Let f :A ----> A be the function f = {(a,c),(b,c),(c,c)}, and let g: A ----> A be the function g = {(a,a),(b,b),(c,a)}. Find g o f and f o g. d. Consider the functions f,g: Z x Z ----> Z x Z defined as f(m,n) = (3m - 4n,2m + n)and g(m,n) = (5m + n,m). Find the formulas for g o f and f o g.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section: Chapter Questions
Problem 63RE
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Question
Prove or solve the following statements:
a. Prove the function f :R - {1} ----> R - {1} defined by f(x) = [(x+1)/(x-1)]^3 is bijective.
b. This question concerns functions f : {A,B,C,D,E} ----> {1,2,3,4,5,6,7}. How many
such functions are there? How many of these functions are injective? How many are surjective? How many are bijective?
c. Suppose A = {a,b,c}. Let f :A ----> A be the function f = {(a,c),(b,c),(c,c)}, and let g: A ----> A be the function g = {(a,a),(b,b),(c,a)}. Find g o f and f o g.
d. Consider the functions f,g: Z x Z ----> Z x Z defined as f(m,n) = (3m - 4n,2m + n)and g(m,n) = (5m + n,m). Find the formulas for g o f and f o g.
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