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renyikouniao
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- Jun 1, 2013
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If a<-1 show that f(x)=ax+cosx and g(x)=ax+sinx are invertible functions;(What are their domain of definitions and ranges?)
How to demonstrate that?Can you demonstrate that for $a<-1$ both functions are monotonic, thus invertible?
For all x<y,f(x)<f(y)?What condition must hold in order for a function to be monotonic?
f'(x)<0 or f'(x)>0What is true about a function's derivative if it is monotonic?
Good, yes, this is what is required for strict monotonicity. As long as the derivative has no roots of odd multiplicity, then the function is monotonic.
Can you show then that for $a<-1$ that the derivatives of the two functions will never change sign?