# Thread: Largest domain and image of a function

1. ## Largest domain and image of a function

Not holding out much hope for this since I can't explain all too well what my issue with it is, but here goes...

Find the largest doman $\displaystyle D\subset R$ (R is the real numbers, can't find the symbol in Latex) such that the rule makes sense and computre the image of $\displaystyle D$ under this rule.

$\displaystyle x\mapsto e^{\tan{x}}$

Now I know $\displaystyle x$ can't equal an odd number $\displaystyle \times\frac{\pi}{2}$, so $\displaystyle x$ (not equal to) $\displaystyle (2n-1)\frac{\pi}{2}, n\epsilon Z$ where $\displaystyle Z$ is the integers (can't find the "not equal to" symbol on the PDF, either).
So for my largest domain I have $\displaystyle \{x|x$ (not equal to) $\displaystyle (2n-1)\frac{\pi}{2}, n\epsilon Z\}$
I'm having trouble getting the set $\displaystyle D$ for which that rule makes sense...

This isn't for marked homework by the way. Well, the theory is, but this examples subtly different from the one I have to hand in to be marked.

2. Originally Posted by chella182
Not holding out much hope for this since I can't explain all too well what my issue with it is, but here goes...

Find the largest doman $\displaystyle D\subset R$ (R is the real numbers, can't find the symbol in Latex) such that the rule makes sense and computre the image of $\displaystyle D$ under this rule.

$\displaystyle x\mapsto e^{\tan{x}}$

Now I know $\displaystyle x$ can't equal an odd number $\displaystyle \times\frac{\pi}{2}$, so $\displaystyle x$ (not equal to) $\displaystyle (2n-1)\frac{\pi}{2}, n\epsilon Z$ where $\displaystyle Z$ is the integers (can't find the "not equal to" symbol on the PDF, either).
So for my largest domain I have $\displaystyle \{x|x$ (not equal to) $\displaystyle (2n-1)\frac{\pi}{2}, n\epsilon Z\}$
I'm having trouble getting the set $\displaystyle D$ for which that rule makes sense...

This isn't for marked homework by the way. Well, the theory is, but this examples subtly different from the one I have to hand in to be marked.
As you noted the domain of $\displaystyle x\mapsto e^{\tan(x)}$ should be $\displaystyle \text{Dom} \text{ }e^x\cap\text{Dom}\text{ }\tan(x)$ and since $\displaystyle \text{Dom}\text{ }e^x=\mathbb{R}$ (\mathbb{R} ) we see that $\displaystyle \text{Dom}\text{ }e^{\tan(x)}=\text{Dom}\text{ }\tan(x)$. Now to find the image (range in this case) try fnind $\displaystyle \text{Dom}\text{ }\text{arctan}\left(\ln(x)\right)$. Or note that $\displaystyle \tan(x)$ is actually able to assume all real values so what values can $\displaystyle e^{\tan(x)}$ assume? Maybe just $\displaystyle \text{Im }e^x$? What do you think?

3. Originally Posted by Drexel28
As you noted the domain of $\displaystyle x\mapsto e^{\tan(x)}$ should be $\displaystyle \text{Dom} \text{ }e^x\cap\text{Dom}\text{ }\tan(x)$ and since $\displaystyle \text{Dom}\text{ }e^x=\mathbb{R}$ (\mathbb{R} ) we see that $\displaystyle \text{Dom}\text{ }e^{\tan(x)}=\text{Dom}\text{ }\tan(x)$. Now to find the image (range in this case) try fnind $\displaystyle \text{Dom}\text{ }\text{arctan}\left(\ln(x)\right)$. Or note that $\displaystyle \tan(x)$ is actually able to assume all real values so what values can $\displaystyle e^{\tan(x)}$ assume? Maybe just $\displaystyle \text{Im }e^x$? What do you think?
Cheers for the Latex tip I think I'm with it now... I just hate stuff like this, it's not what I'm good at