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Math Help - Domain and range on university level

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    Domain and range on university level

    Hi, just finished my first calculus lesson in university. I am just wondering how does one express the range in university notation? I have this question, express the range of the function of y = x^(-1). I have no idea how to right it on the university level. I would write it as yeR, y =/= 0. How would I write it in university level because the teacher claims not all numbers are going to be real that is why you use the notation of "D".

    Also, how do I generalize the domain of y=tan(x) in university level? I ask this because I have no idea how to show the asymptotes mathematically.
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    Re: Domain and range on university level

    Admitedly, I don't know what "university notation" is. This site University Maths Notes - Set Theory - Notation for Set Theory
    has examples of interval notation...

    For the first example, the range of y = x^(-1) = 1/x is all real numbers except zero. So you could write
    D = ℝ \ {0}, or
    D = {y : y ∈ ℝ and y =/= 0}, or
    D = (-∞, 0) U (0, ∞),
    depending on the prof.
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    Re: Domain and range on university level

    The tangent function has vertical asymptotes at odd multiplies of \frac{\pi}{2}. In "interval notation" that would be written as \cup_{n=-\infty}^\infty \left(\frac{(2n-1)\pi}{2}, \frac{(2n+1)\pi}{2}\right), the union of all intervals between such numbers.

    In "set builder notation" it would be \{x\in R | x\ne \frac{(2n+1)\pi}{2}, n\in Z\}, the set of all real numbers except odd multiples of \pi/2
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    Re: Domain and range on university level

    Quote Originally Posted by HallsofIvy View Post
    The tangent function has vertical asymptotes at odd multiplies of \frac{\pi}{2}. In "interval notation" that would be written as \cup_{n=-\infty}^\infty \left(\frac{(2n-1)\pi}{2}, \frac{(2n+1)\pi}{2}\right), the union of all intervals between such numbers.

    In "set builder notation" it would be \{x\in R | x\ne \frac{(2n+1)\pi}{2}, n\in Z\}, the set of all real numbers except odd multiples of \pi/2
    Why is it that the number factor cannot be odd and can I multiply the entire faction by two to get rid of the bottom two? Like so, 2pi(2n+1)?
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    Re: Domain and range on university level

    Quote Originally Posted by Barthayn View Post
    Why is it that the number factor cannot be odd and can I multiply the entire faction by two to get rid of the bottom two? Like so, 2pi(2n+1)?
    Remember that \displaystyle \tan{x} = \frac{\sin{x}}{\cos{x}} , and is defined wherever the denominator is nonzero. Where is the denominator zero?
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