Discrete Mathematics with Graph Theory (3rd Edition) 70

Discrete Mathematics with Graph Theory (3rd Edition) 70 -...

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68 Solutions to Exercises (b) [BB] The function f: (2,5) - (0,1) defined by f(x) = ix - ~ is a one-to-one correspondence, as is the function g: (0,1) - (10,00) defined by g(x) = .!. + 9. Thus, the composition go x f: (2,5) - (10,00) is also a one-to-one correspondence. Note that (g 0 f)(x) = ~2 + 9. x- (c) The function f: (a, b) - (0,1) defined by f(x) = xb - a is a one-to-one correspondence, as is -a the function g: (0, 1) - (c, 00) defined by g(x) = .!. - 1 + c. Thus a one-to-one correspondence x (a, b) _ (c, 00) is the composition 9 0 f. Note that (g 0 f)(x) = b - a - 1 + c. x-a 14. In Problem 27, p. 129, we saw that the function g: (0,1) - (0,00) defined by g(x) = .!. - 1 is a x one-to-one correspondence. Furthermore, the function f: (a, b) _ (0,1) defined by f(x) = xb - a is -a a one-to-one correspondence. (See the remarks preceding Problem 28.) Thus 9 0 f: (a, b) - (0,00) b-a is a one-to-one correspondence. Note that (g 0 f)(x) = -- -l. x-a
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