Discrete Mathematics with Graph Theory (3rd Edition) 65

Discrete Mathematics with Graph Theory (3rd Edition) 65 - 9...

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Section 3.2 63 15. (f 0 9 )( x) = I (g (x» = I (x - c). Thus the graph of log is the graph of I, but translated horizontally c units to the right if c > 0 and -c units to the left if c < O. The graphs are identical if c = O. 16. (a) [BB] Since -Ixl = {-x ~f x ~ 0 we have I 0 g(x) = I( -Ixl) = {/((-)X) ~f x ~ 0 x 1f x < 0 I x 1f x < O. So the graph of log is the same as the graph of I to the left of the y-axis (where x < 0) while to the right of the y-axis, the graph of log is the reflection (mirror image) of the left half of the graph of I in the y-axis. We call log an even function since it is symmetric with respect to the y-axis: 1 0 g( -x) = 10 g(x). { - I(x) if I(x) ~ 0 (b) Sincego/(x) = -1/(x)1 = () . () thegraphofgo/isthesameasthegraph Ix 1flx <0 of I wherever the graph of I is below the x-axis (y < 0) while it is the reflection (mirror image) of the graph of I in the x-axis wherever the graph of I is above the x-axis. (In particular, the graph of
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Unformatted text preview: 9 0 I lies entirely on or below the x-axis.) 1 17. (a) 1 0 g(x) = I(g(x» = I( 1~:zJ = 1--1-= 1- (1 -x) = £(x). 1-x 1 go r(x) = g(r(x» = g(x:'1) = 1-. ...L = (x:'l{-x = X:: 1 1 = 1-x = s(x). x-1 1 11£ I 9 h r sl £ £ I· 9 h r s I I 9 £ S h r 9 9 £ I r s h h h r s £ I 9 r r s h 9 £ I s s h r I 9 £ inverse (b) All these functions have inverses. function 18. (a) 0 £ It h fa 14 15 £ £ It h fa 14 15 It It £ 14 15 h fa h h 15 £ !4 fa h fa fa !4 15 £ h h 14 !4 fa It h 15 £ 15 15 h fa 11 £ 14 (b) All these functions have inverses. inverse function 19. (a) [BB] log = {(1, 4), (2,3), (3,2), (4, 1), (5, 5)}; 9 0 1= {(1, 5), (2,3), (3,2), (4,4), (5, 1)} Clearly, log =I 9 0 I. (b) 1-1 = {(1, 2), (2, 1), (3, 5), (4,3), (5, 4)}; g-1 = {(1, 3), (2,4), (3, 1), (4,5), (5, 2)} Functions I and 9 have inverses because they are one-to-one and onto while h does not have an inverse because it is not one-to-one (equally because it is not onto)....
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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.

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