MHB Finding domain of a trigonometric function

[tmt1]

I need to find the domain of this function:

$$f(x) = \frac{1+x}{ e^{cos(x)}}$$

I set $${ e^{cos(x)}}> 0$$

But I'm not sure what to do after this.

 

[Prove It]

I need to find the domain of this function:

$$f(x) = \frac{1+x}{ e^{cos(x)}}$$

I set $${ e^{cos(x)}}> 0$$

But I'm not sure what to do after this.

The top is obviously defined for all x.

The bottom, as you have established, is always positive. As both cos(x) and e^x are defined for all x, so is their composition.

So the top and bottom are defined for all x. The only place where the quotient of the two functions might not be defined is where the denominator is 0. But you already established that this doesn't happen anywhere.

So what is the domain of your function?

 

[tmt1]

The top is obviously defined for all x.

The bottom, as you have established, is always positive. As both cos(x) and e^x are defined for all x, so is their composition.

So the top and bottom are defined for all x. The only place where the quotient of the two functions might not be defined is where the denominator is 0. But you already established that this doesn't happen anywhere.

So what is the domain of your function?

All real numbers?

 

[Prove It]

All real numbers?

Correct :)