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Discrete Mathematics with Graph Theory (3rd Edition) 62

# Discrete Mathematics with Graph Theory (3rd Edition) 62 -...

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60 Solutions to Exercises 4. Let y = f-l(x). Then x = f(y) = -.;y, so y = x 2 = f-l(X). 5. Since g(x) is an integer, f o. g(x) = g(x). Similarly, 9 0 f(x) = f(x). 6. (a) [BB] f- l : R -+ R is defined by f-i(x) = i(x - 5). (b) f- l : R -+ R is defined by f-l(X) =(x + 2)1/3. (c) {3-l: R -+ (~, 00) is given by (3-l(x) =~(2x.+ 4) . . {v'x ifx~O (d) g-l: R ~ R is defined by g-l(X) = . .. _. r-=_x Y-'" if x < o. (a) (b) y 3 -2 -1 -3 -4 (c)y (d) 6 1 1 1 1 7. (a) [BB] If f(xt) = f(X2), then 1 + --4 = 1 + --4' --4 = --4 and Xl - 4 = X2 - 4. Xl - X2 - Xl - X2 - Thus Xl = X2 and f is one-to-one. Next
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Unformatted text preview: .y E rngj ~ y == f(:1;) for some x EA ~ there is an X E.A such that y = 1 + ~4 x-.' 1 +-+ there is an X E A such that y -1 = X _ 4 +-+ there is an X E A such that (y ~ l)(x - 4) = 1 +-+y#L Thus rng f = B = {y E R I y # I} .and f has an inverse B -+ A, To find a formula f~r f-l(x), 1 1 let y = f-l(x), x E B. Then x = f(y) = 1 + --4' so x-I = --4' (x -l)(y -4) = 1 . y- y-and, since x # 1, y - 4 = ~1 and.J:-l(x) = y = 4 + ~1. x-· x-l 1 1 1 (b) Supposef(xl) = f(X2). Then5--1--= 5--1--,so -1--= -1--' l+Xl = 1+x2 . + Xi + X2 + Xl + X2...
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