domain of a function


I need help finding the domain of the function


x < 3sqrt(10) && x != sqrt(90-18e-e^2) ??


I assume you're just solving this by hand and not asking how to do this with a 49G+/other graphing model.

Remember that the domain of a function is simply the set of x values for which the function is defined. To find where the function is not defined in this case you must find solutions that result in 1/0. If you set the denominator equal to 0 and solve for x you will find the domain. To get you started:


Some relatively basic algebra will give you the answer. Keep the properties of the squared term in mind. Good luck!


Hi, John;

Hugh is correct, one of the first restrictions is the one he pointed out. Now, let me add some guidance that I hope will help you a bit. When defining/checking domain, discontinuities, poles and, if applicable, undefined points must be found so you can set all conditions that define your domain. For each particular function you'll have a series of restrictions. I guess that you may find your way to get the answer yourself, just take some guidelines for this particular case:

1 - denominator different of zero, meaning:

1-ln(9-sqrrt(x^2-9)) =! 0 ,hence
ln(9-sqrt(x^2-9)) =! 1
9-sqrt(x^2-9) =! e (2.7183...)
sqrt(x^2-9) =! 9-e (now you isolate x... your turn!)
2 - because it is a ln operand:
9-sqrt(x^2-9) > 0  , hence:
sqrt(x^2-9) > 9
3 - sqrt parameter should be greater than zero as well:
x^2-9  >  0  , hence
x^2 > 9 and
|x| > 3
Can you put all of these conditions together in one only? If so, you'll have your domain.

(at least I hope this is correct; have someone found anythng wrong here, please add corrections)


Luiz (Brazil)


Well put, Luiz. I was somewhat hesitant to elucidate since this looks to be a homework problem. You did an excellent job of laying the logical groundwork and demonstrating optimal technique to determine the solution without giving much away.



Hi, Hugh; thanks for your comments.

You see, that's the technique I always try to use when teaching: "teasing" to an answer. Not always possible, depends on the problem itself and the subject, but when applicable, it is rewarding for both teacher and students. I, as a teacher, must "feel" where to stop (sometimes going further, sometimes stopping earlier), and the students, in their turn, must feel confident going ahead by their own.

I don't remember attending classes like the ones I propose to my students today, but I guess we must all evolve, enhance. Your comments show me I'm on the right path. I neither blame my teachers for their classes, I am confident they were doing the best they could. You see, I tell my students that when the job is done by both teacher and students, learning is a consequence of teaching and examination is just a matter of bureaucratic evidence. If students search for knowledge and do not feel satisfied until their doubts are gone, (good) teaching is always effective.


Luiz (Brazil)

Edited: 8 Feb 2005, 11:41 p.m.

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