Largest domain and image of a function

**chella182** · November 9th 2009, 06:15 AM

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

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

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

**Drexel28** · November 9th 2009, 06:31 AM

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

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

Now I know $x$ can't equal an odd number $\times\frac{\pi}{2}$ , so $x$ (not equal to) $(2n-1)\frac{\pi}{2}, n\epsilon Z$ where $Z$ is the integers (can't find the "not equal to" symbol on the PDF, either).
So for my largest domain I have $\{x|x$ (not equal to) $(2n-1)\frac{\pi}{2}, n\epsilon Z\}$
I'm having trouble getting the set $D$ 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 $x\mapsto e^{\tan(x)}$ should be $\text{Dom} \text{ }e^x\cap\text{Dom}\text{ }\tan(x)$ and since $\text{Dom}\text{ }e^x=\mathbb{R}$ (\mathbb{R}

) we see that $\text{Dom}\text{ }e^{\tan(x)}=\text{Dom}\text{ }\tan(x)$ . Now to find the image (range in this case) try fnind $\text{Dom}\text{ }\text{arctan}\left(\ln(x)\right)$ . Or note that $\tan(x)$ is actually able to assume all real values so what values can $e^{\tan(x)}$ assume? Maybe just $\text{Im }e^x$ ? What do you think?

**chella182** · November 9th 2009, 06:45 AM

Originally Posted by Drexel28

As you noted the domain of $x\mapsto e^{\tan(x)}$ should be $\text{Dom} \text{ }e^x\cap\text{Dom}\text{ }\tan(x)$ and since $\text{Dom}\text{ }e^x=\mathbb{R}$ (\mathbb{R}

) we see that $\text{Dom}\text{ }e^{\tan(x)}=\text{Dom}\text{ }\tan(x)$ . Now to find the image (range in this case) try fnind $\text{Dom}\text{ }\text{arctan}\left(\ln(x)\right)$ . Or note that $\tan(x)$ is actually able to assume all real values so what values can $e^{\tan(x)}$ assume? Maybe just $\text{Im }e^x$ ? 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

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