(a) Define F: Z → Z by the rule F(n) = 2 - 3n, for each integer n. n1 and n2 are any integers, such that F(n₁) = F(n2). Substituting from the definition of F gives that 2 - 3n₁ = = Solving this equation for n₁ and simplifying the result gives that n₁ = Therefore, F is (i) Is F one-to-one? Suppose ---Select--- (ii) Show that F is not onto. Counterexample: Let m = 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 = (1) Is x a real number? Sums, products, and differences of real numbers are --Select--- (2) Does G(x) = y? and quotients of real numbers with nonzero denominators are --Select--- . Therefore, x ---Select--- 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---