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Invertible functions Question 1

renyikouniao

Member
Jun 1, 2013
41
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?)
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Can you demonstrate that for $a<-1$ both functions are monotonic, thus invertible?
 

renyikouniao

Member
Jun 1, 2013
41

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
What condition must hold in order for a function to be monotonic?
 

renyikouniao

Member
Jun 1, 2013
41
What condition must hold in order for a function to be monotonic?
For all x<y,f(x)<f(y)?
Or for all x>y,f(x)<f(y)
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
What is true about a function's derivative if it is monotonic?
 

renyikouniao

Member
Jun 1, 2013
41

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
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?
 

renyikouniao

Member
Jun 1, 2013
41
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?

like using the graph?
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
I would do it algebraically. For the first function, we are given

\(\displaystyle f(x)=ax+\cos(x)\)

Differentiating, we find:

\(\displaystyle f'(x)=a-\sin(x)\)

Next, I would begin with:

\(\displaystyle -1\le-\sin(x)\le1\)

Can you get $f'(x)$ in the middle, and then use $a<-1$ to show that $f'(x)<0$?