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Math Help - Largest domain and image of a function

  1. #1
    Senior Member chella182's Avatar
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    Exclamation 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 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.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by chella182 View Post
    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?
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  3. #3
    Senior Member chella182's Avatar
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    Quote Originally Posted by Drexel28 View Post
    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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