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Find the domain of the function.

$ g(x) = \dfrac{1}{1 - \tan x} $

$x \neq \frac{\pi}{2}+\pi n$ , $x \neq \frac{\pi}{4}+\pi n$

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Johns Hopkins University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

to find the domain of this function. There's a couple different things we want to think about. First of all, Tangent itself has some domain restrictions. So if we were just looking at the tangent function, why equals tangent? X X could not be hi over to Tangent has a vertical Assen tote there three pi over to another vertical Lassen tote five pi over two, etcetera. There's infinitely many of them on the way we can consolidate. How we express that is we could say that X cannot equal pi over two plus pi times in where in is an integer so and could be one and could be too, and could be three. And we're just adding more multiples of pi and getting more answers. So that's one part of the domain restriction for this function. The other part is that the entire denominator cannot equal zero because if the denominator is zero, you haven't undefined expression. So we have to figure out what makes one minus tangent X equal to zero. So if I saw this equation, I get tangent X equals one, and then I get X is the inverse tangent of one, some thinking about angles that have a tangent value of one. And if I picture the unit circle or a reference triangle, if I have a 45 degree reference angle, then I'm going to have sides of length one and one and square root to so 45 degrees has a tangent value of one and 45 degrees is equivalent to pi over four radiance. Now that will also happen in Quadrant three. And so we're gonna have an answer of 225 degrees, which is equivalent to five pi over four radiance. And it happens over and over again as you keep going around two more angles that are co terminal with these two. So they're infinitely many answers. So we end up with exes pi over four. Add another pie to it. You get five pi over four at another pie to it, you get nine pi over four. So pi over four plus pi times and is how well express that. So what we're saying is this function cannot have any of those answers, and it cannot have any of the previously described answers. So we can just say that the domain of this function is X is an element of the rial numbers such that X is not equal to pi over two plus pi. Times in or X is not equal to pi over four plus pi in and I should say, and instead of or to make sure that both of these air true Okay, there's very complicated looking domain.