Define F: Z → Z by the rule F(n) = 2 − 3n, for each integer n. (i) Is F one-to-one? Suppose n1 and n2 are any integers, such that F(n1) = F(n2). Substituting from the definition of F gives that 2 − 3n1 = 2−3n2​ . Solving this equation for n1 and simplifying the result gives that n1 = 2−3n2​ . Therefore, F is one-to-one . (ii) Show that F is not onto. Counterexample: Let m = n−23 . For this value of m, the only number n with the property that F(n) = m is not an integer. Thus, F is not onto. (b) Define G: R → R by the rule G(x) = 2 − 3x for each real number x. Is G onto? Scratch work: Let y be any real number. On a separate piece of paper, solve the equation y = 2 − 3x for x. Enter the result—an expression in y—in the box below. x = y (1) Is x a real number? Sums, products, and differences of real numbers are ---Select--- , and quotients of real numbers with nonzero denominators are ---Select--- . Therefore, x ---Select--- . (2) Does G(y) = x? According to the formula that defines G, when G is applied to x, x is multiplied by 3 and the result is subtracted from 2. When the expression for x that you found above is multiplied by 3, the result is . And when the result is subtracted from 2, you obtain . Thus, ---Select--- . Hence, ---Select--- a number x such that x is a real number and G(x) = y. Therefore, ---Select--- .