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 :)

 

[CellCoree]

As a scientist, it is important to understand the domain of a function in order to properly analyze and interpret its behavior. In this case, the domain of the function f(x) can be determined by considering the restrictions on the input values that would result in a meaningful output.

Since the function involves a trigonometric function (cos(x)) and an exponential function (e^x), we need to consider the restrictions on the input values for both of these functions. The cosine function has a domain of all real numbers, while the exponential function has a domain of all real numbers as well. However, the output of the exponential function is always positive, meaning that the fraction in the function f(x) will only be defined for values of x that result in a positive output for the exponential function.

Therefore, the domain of the function f(x) can be written as:
$$
\boxed{ \{ x \in \mathbb{R} \mid e^{cos(x)} > 0 \} }
$$

In other words, the domain of the function f(x) is all real numbers except for the values of x that result in a negative or zero output for the exponential function, which would make the fraction undefined. This can also be written as:
$$
\boxed{ \{ x \in \mathbb{R} \mid cos(x) \neq ln(0) \} }
$$

Overall, understanding the domain of a function is crucial for analyzing its behavior and making accurate conclusions. By considering the restrictions on the input values for each component of the function, we can determine the valid domain for the function f(x).