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.
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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text: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 x.
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_ƒf(x) = -∞.
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 x, so (fog)(y) = y for every y
in the range of f.
5. Steps 2 and 4 imply that fog is the identity function.
Which one of the following statements best describes this proposed proof?
Step 1 is faulty.
The proof is valid.
Step 5 is faulty.
Step 4 is faulty.
Step 3 is faulty.
Step 2 is faulty.
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