1. For all real x, let the functions f, g, and h be defined, on the domain of all real numbers, as follows:
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f(x) = (x3/3) + (x2/2) + x + 1 |
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g(x) = x3 + x2 + x + 1 |
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h(x) = (x3/3) + (5x2/2) + 6x + 1 |
Which of these functions has an inverse?
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A. f and g only |
B. f and h only |
C. g and h only |
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D. f, g, and h |
E. The correct answer is not given by A, B, C, or D. |
Solution: The answer is A
By definition, let f be a one-to-one function with domain A and range B. Then the inverse function f-1 has domain B and range A, defined by,
for any y in B.
To find the inverse functions of the given functions above is not trivial.But, by a theorem, all increasing and all decreasing functions are one-to-one functions, so have an inverse.
The derivatives of the given functions are:
Since f’(x) > 0 and g’(x) > 0 for all real numbers x, then f(x) and g(x) are increasing functions and thus one-to-one functions.
Notice that h(x) has a local extremum (that is, maximum or minimum) at x = -3 and x = -2 which implies that h(x) is not a one-to-one function since there exist two different values of x that will yield the same value for h(x).