# Invertible functions Question 1

#### renyikouniao

##### Member
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

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

#### renyikouniao

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

#### MarkFL

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

#### renyikouniao

##### Member
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

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

#### renyikouniao

##### Member
What is true about a function's derivative if it is monotonic?
f'(x)<0 or f'(x)>0

#### MarkFL

Staff member
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
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

Staff member
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 $f'(x)<0$?