The domain of the function $$f(x) = \frac{1+x}{ e^{cos(x)}}$$ is all real numbers. The numerator, $$1+x$$, is defined for all x. The denominator, $$e^{cos(x)}$$, is always positive since both $$cos(x)$$ and $$e^x$$ are defined for all x, ensuring that the composition is also defined everywhere. Therefore, there are no restrictions on the domain, confirming that the function is defined for all real numbers.
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tmt said: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 said: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?
tmt said:All real numbers?