MHB Are f(x)=ax+cos(x) and g(x)=ax+sin(x) Invertible for a<-1?

[renyikouniao]

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]

Can you demonstrate that for $a<-1$ both functions are monotonic, thus invertible?

 

[renyikouniao]

Can you demonstrate that for $a<-1$ both functions are monotonic, thus invertible?

How to demonstrate that?

 

[MarkFL]

What condition must hold in order for a function to be monotonic?

 

[renyikouniao]

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]

What is true about a function's derivative if it is monotonic?

 

[renyikouniao]

What is true about a function's derivative if it is monotonic?

f'(x)<0 or f'(x)>0

 

[MarkFL]

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]

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]

I would do it algebraically. For the first function, we are given

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

Differentiating, we find:

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

Next, I would begin with:

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

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