aug23 - Math 3124 Tuesday, August 23 August 23 Ungraded...

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Math 3124 Tuesday, August 23 August 23 Ungraded Homework Problem 1.2 on page 14 Let α , β and γ be mappings from Z to Z defined by α ( n ) = 2 n , β ( n ) = n + 1, and γ ( n ) = n 3 for each n Z . (a) Which of the three mappings are onto? (b) Which of the three mappings are one-to-one? (c) Determine α ( N ) , β ( N ) , and γ ( N ) . (a) β is onto, the others are not. (b) α , β , γ are all one-to-one. (c) α ( N ) = positive even integers { 2 , 4 ,... } . β ( N ) = integers 2 { 2 , 3 ,... } . γ ( N ) = all cubes { 1 , 8 , 27 ,... } . Problem 1.14 on page 14 Define f : R R by f ( x ) = x 2 + x . Determine whether f is onto, and whether f is one-to-one. Also describe f ( P ) where P is the positive real numbers. We are given f : R R defined by f ( x ) = x 2 + x . This is not onto; indeed by calculus the minimum of f occurs when f 0 ( x ) = 0, that is when 2 x + 1 = 0, which is when x = - 1 / 2 and then f ( x ) = 1 / 4 - 1 / 2 = - 1 / 4. Thus f ( x ) ≥ - 1 / 4 for all
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aug23 - Math 3124 Tuesday, August 23 August 23 Ungraded...

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