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.

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.

Re: Domain and range on university level

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

In "set builder notation" it would be $\displaystyle \{x\in R | x\ne \frac{(2n+1)\pi}{2}, n\in Z\}$, the set of all real numbers **except** odd multiples of $\displaystyle \pi/2$

Re: Domain and range on university level

Quote:

Originally Posted by

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

In "set builder notation" it would be $\displaystyle \{x\in R | x\ne \frac{(2n+1)\pi}{2}, n\in Z\}$, the set of all real numbers **except** odd multiples of $\displaystyle \pi/2$

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)?

Re: Domain and range on university level

Quote:

Originally Posted by

**Barthayn** 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 \displaystyle \tan{x} = \frac{\sin{x}}{\cos{x}} $, and is defined wherever the denominator is nonzero. Where is the denominator zero?