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11.09 - Chapter Review true/false questions on page...

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Chapter Review true/false questions on page 195 #1: “If f is one-to-one, with domain R , then f –1 ( f (6)) = 6.” ..?.. Answer: True. But note that f ( f –1 (6)) need not be defined; e.g., let f ( x ) = – exp( x ) or exp( x ) + 6. #2: “If f is one-to-one and differentiable, with domain R , then ( f –1 ) (6) = 1/ f (6).” ..?.. Answer: False. It’s equal to 1/ f ( f –1 (6)) (assuming that the denominator isn’t zero), and it’s easy to cook up examples like f ( x ) = x 3 where the two quantities are unequal. #3: “The function f ( x ) = cos x , – π /2 x π /2, is one-to- one.” ..?.. Answer: False (note for instance that cos π /2 = cos – π /2). But f ( x ) = cos x , 0 x π is one-to-one, as is f ( x ) = sin x , π /2 x π /2. #4: “tan –1 (–1) = 3 π /4” ..?.. Answer: False! tan(3 π /4) = –1, but since the range of tan –1
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is (- π /2, π /2) we must have tan –1 (–1) = – π /4. #5: “If 0 < a < b , then ln a < ln b .” ..?.. Answer: True, since ln x is an increasing function on (0, ).
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