Question

Define G: ℝ → ℝ by the rule G(x) = 2 − 3x for each real...

Define G: ℝ → ℝ by the rule G(x) = 2 − 3x for each real number x. Prove that G is onto.

Proof that G is onto: Let y be any real number. (Scratch work: 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=________

To finish the proof, we need to show (1) that x is a real number, and (2) that G(x) = y.

Now sums, products, and differences of real numbers are real numbers, and quotients of real numbers with nonzero denominators are also real numbers. Therefore, x is a real number.

In addition, 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 (using the variable y) is multiplied by 3, the result is ______. And when the result is subtracted from 2, we obtain ______. Thus G(x) = y. Hence, there exists a number x such that x is a real number and G(x) = y. Therefore, G is onto.

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