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

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Discussion Overview

The discussion centers on the invertibility of the functions f(x) = ax + cos(x) and g(x) = ax + sin(x) for the condition a < -1. Participants explore the conditions under which these functions are monotonic, which is a key aspect of their invertibility, and consider the implications of their derivatives.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants inquire about demonstrating that both functions are monotonic for a < -1, which would imply they are invertible.
  • There is a discussion on the conditions required for a function to be monotonic, with questions about whether it holds that for all x < y, f(x) < f(y) or for all x > y, f(x) < f(y).
  • Participants discuss the significance of a function's derivative in determining monotonicity, noting that f'(x) must be either always less than or always greater than zero.
  • One participant suggests that as long as the derivative has no roots of odd multiplicity, the function will be monotonic.
  • There is a proposal to show that for a < -1, the derivatives of the two functions will never change sign, potentially using graphical methods or algebraic approaches.
  • Another participant begins to differentiate f(x) and considers the implications of the derivative f'(x) = a - sin(x) in relation to the condition a < -1.

Areas of Agreement / Disagreement

Participants generally agree on the need to establish monotonicity for invertibility, but the methods and specific conditions required for demonstrating this remain under discussion. There is no consensus on the best approach to show that the derivatives do not change sign.

Contextual Notes

Limitations include the need for further clarification on the behavior of the derivatives and the specific implications of the condition a < -1 on the functions' monotonicity.

renyikouniao
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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?)
 
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Can you demonstrate that for $a<-1$ both functions are monotonic, thus invertible?
 
MarkFL said:
Can you demonstrate that for $a<-1$ both functions are monotonic, thus invertible?

How to demonstrate that?
 
What condition must hold in order for a function to be monotonic?
 
MarkFL said:
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)
 
What is true about a function's derivative if it is monotonic?
 
MarkFL said:
What is true about a function's derivative if it is monotonic?

f'(x)<0 or f'(x)>0
 
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?
 
MarkFL said:
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?
 
  • #10
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 $f'(x)<0$?
 

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