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Question: find the domain of the following function $g(x) = (x^2 - 6x)^{1/4}$

My try:

Since the expression $x^2 - 6x$ is under an even root, then the expression should be greater than or equal to zero, hence:

$x^2 - 6x \geq 0$

$\implies x(x - 6) \geq 0$

$\implies x \geq 0$ or $x \geq 6$

What am I doing wrong?

The solution given in the website.

A similar problem

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  • $\begingroup$ look at page 13 of this arihantbooks.com/chapterdownloads/B015.pdf $\endgroup$ May 2, 2016 at 6:46
  • $\begingroup$ It is sometimes easier to solve these problems by visualizing the graph rather than purely using algebra. How does the graph of $f(x) = x^2 - 6x$ look like? It is a concave-up parabola with zeroes at $x=0$ and $x=6$. The answer should be clear to you then. $\endgroup$ May 2, 2016 at 6:48

1 Answer 1

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Inequalities don't factor like that.

$a*b\geq 0$ does not mean ($a\geq 0$ or $b\geq 0$).

It means (($a\geq0$ and $b\geq0$) or ($a\leq 0$ and $b\leq 0$)).

Because two positive numbers multiplied together give a positive number, and two negative numbers multiplied together also give a positive number.

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  • $\begingroup$ So, with that logic, I go for the 1st one and my answer should be correct. $\endgroup$ May 2, 2016 at 6:46
  • $\begingroup$ That is a mathematical "or", inclusive or. You are asked to find the full domain, not just part of it. $\endgroup$
    – Wouter
    May 2, 2016 at 6:49
  • $\begingroup$ Ah, okay, got it. So, the answer really would be $x \geq 6$ and $x \geq 0$ or $x \leq 6$ and $x \leq 0$. Right? $\endgroup$ May 2, 2016 at 6:51
  • $\begingroup$ Yes. That simplifies to $x\geq 6$ or $x\leq 0$, the answer given in your textbook. $\endgroup$
    – Wouter
    May 2, 2016 at 6:54

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