Discrete Mathematics with Graph Theory (3rd Edition) 57

Discrete Mathematics with Graph Theory (3rd Edition) 57 -...

This preview shows page 1. Sign up to view the full content.

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 = - 13 4 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.
This is the end of the preview. Sign up to access the rest of the document.

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.

Ask a homework question - tutors are online