Increasing Inverse
Given that is a one-to-one, strictly increasing function. Is it necessary true that its inverse function is also a strictly increasing function?
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1 solution
If its inverse function isn't strictly increasing function, then it exist two numbers a , b such that a > b and f − 1 ( a ) ≤ f − 1 ( b ) .
As f ( x ) is strictly increasing function, and f − 1 ( a ) ≤ f − 1 ( b ) . We get f ( f − 1 ( a ) ) ≤ f ( f − 1 ( b ) ) , which is equivalent to a ≤ b . Which is contradiction. Hence, the answer is t r u e .