Section 3.1 55 17. (a) The function is not onto since, for example, g(x) = 0 has no integer solutions (by the quadratic formula), but it is one-to-one. To see why, suppose g(X1) = g(X2). Then 3x~ + 14x1 -51 = 3x~ + 14x2 - 51, so 3(X1 -X2)(X1 + X2) = 14(x2 -Xl). Thus Xl = X2 or Xl + X2 = -li. Since Xl and X2 are integers, Xl + X2 = - 134 is impossible. Thus, Xl = X2, so 9 is one-to-one. (b) Since the graph of 9 is a parabola, 9 is neither one-to-one nor onto. 18. (a) [BB] Note that f(x) = (x + 7)2 -100. If f(X1) = f(X2) it follows that (Xl + 7)2 -100 = (X2 + 7)2 -100, so (Xl + 7)2 = (X2 + 7)2 and, taking square roots, IX1 + 71 = IX2 + 71. Since Xl, X2 E N, we know Xl + 7 > 0 and X2 + 7 > O. Thus, Xl + 7 = X2 + 7 and Xl = X2. Thus, f is one-to-one, but it is not onto: For example, 1 E B, but there is no X E N with 1 (x) = 1 since (x + 7)2 -100 = 1 implies (x + 7)2 = 101 and this equation has no solution in the natural numbers.
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.