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 October 25th, 2016, 06:19 AM #1 Newbie   Joined: Oct 2016 From: Indonesia Posts: 1 Thanks: 0 find the definition area of the function October 25th, 2016, 07:12 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 "Definition area"? Do you mean the "natural domain" of the function- the region in which the formula is defined? In order that a number have a square root that number must be non-zero. Here, we must have $\displaystyle x- \frac{1}{x}\ge 0$. If x is positive, multiplying both sides by x we have $\displaystyle x^2- 1= (x- 1)(x+ 1)\ge 0$. In order for that to be true, both x- 1 and x+ 1 must have the same sign: either x-1> 0 and x+ 1> 0 or x- 1< 0 and x+ 1< 0. The first pair of inequalities is true for x> 1 and the second for x< -1. Since "x is positive", we must have x> 1. If x is negative, multiplying both sides by x we have $\displaystyle x^2- 1= (x- 1)(x+ 1)\ge 0$. In order for that to be true, x- 1 and x+ 1 must have opposite signs: x+ 1> 0 and x- 1< 0 or x- 1<0 and x- 1> 0. The first pair is true for $\displaystyle -1\le 0 \le 1$. The second pair are never both true. Since "x is negative" we must have $\displaystyle -1\le x\le 0$. Of course, we also cannot divide by 0 so x= 0 is not in the domain. The domain is the union of the two separate sets:$\displaystyle \{ x|-1\le x< 0\}\cup \{ x| x> 1\}$. Last edited by Country Boy; October 25th, 2016 at 07:17 AM. October 25th, 2016, 07:17 AM #3 Senior Member   Joined: May 2016 From: USA Posts: 1,310 Thanks: 551 It is actually impossible to answer this question because it is ambiguous. If you are talking about real functions, there is one answer, and if you are talking about complex functions, there is a different answer. Let's suppose we are talking about real functions of real variables, and let's define $g(x) = x - \dfrac{1}{x}\ and\ f(x) = \sqrt{g(x)}\ such\ that\ x,\ g(x),\ f(x) \in \mathbb R.$ For what values of g(x) is f(x) a real number? Consequently, for what values of x is f(x) a real number? October 25th, 2016, 11:34 PM   #4
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 Originally Posted by noreason Attachment 8048
Hi fellow Indonesian! To be honest, we don't know what you are asking for. Maybe you can post the question in our native language so I can tell the others what do you intend to ask?  Tags area, definition, find, function Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post EvanJ Pre-Calculus 3 October 6th, 2015 01:28 PM rain Calculus 4 November 2nd, 2013 01:45 PM yugimutoshung Algebra 2 April 26th, 2013 09:21 PM safyras Calculus 2 December 5th, 2010 08:25 AM Andrey Calculus 1 February 3rd, 2008 09:45 AM

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