Suppose f: R → R is defined by the property that f(x) = x + x² + x³ for every real number x, and g: R → R has the property that (gof)(x) = x for every real number . Evaluate the following proposed proof that (fog)(x) = x for every real number . 1. Exercise 3.5.6(b) implies that lim f(x) = ∞ and lim f(x) = -∞. 2. Step 1 and the intermediate-value theorem imply that the range of the function f is all of R. 3. If g(f(x)) = x, then f(g(f(x))) = f(x). 4. The conclusion of Step 3 holds for every real number a, so (fog)(y) = y for every y in the range of f. 5. Steps 2 and 4 imply that fog is the identity function.

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