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October 25th, 2016, 06:19 AM   #1
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Post find the definition area of the function

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October 25th, 2016, 07:12 AM   #2
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"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.
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October 25th, 2016, 07:17 AM   #3
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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?
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October 25th, 2016, 11:34 PM   #4
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Originally Posted by noreason View Post
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?
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