Section 7.2:

(a) Define F: Z->Z by the rule F(n) = 2-3n, for all integers n.

(i) is F one-to-one? Prove or give a counterexample.
(ii) Is F onto? Prove or give coutnerexample.

(b) Define G: R->R by the rule G(x) = 2-3x for all real numbers x. Is G onto? Prove or give counterexample.


f(x) = (3x-1)/(x), for all real numbers x != 0


f(x) = (x+1)/(x-1), for all real numbers x != 1

Section 7.3:


Define H: Z->Z and K:Z->Z by the rules H(a) = 6a and K(a) = a mod 4 for all integers a. Find (K o H) (0), (K o H)(1), (K o H)(2), and (K o H) (3).


The functions of each pair are inverse to each other. For each pair, check that both compositions give the identity function. (11)

H and H⁻¹ are both defined from R- {1} to R - {1} by the formula H(x) = H⁻¹(x) = (x+1)/(x-1), for all x IN R - {1}


Prove or give a counterexample: If f: X -> Y and g: Y ->X are functions such that g o f = Iₓ and f o g = I_Y then f and g are both one-to-one and onto and g = f⁻¹.

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