1. ## Finding the domain

The question:
Determine f/g, and give the domain.

$\displaystyle f(x) = \sqrt{x -1}$
$\displaystyle g(x) = x - \sqrt{x + 1}$

I did:

$\displaystyle \frac{\sqrt{x-1}}{x - \sqrt{x + 1}}$

And I found that x had to be >= 1 (since you can't square root a negative value). I wrote [1, inf) as my answer. However, the answer is $\displaystyle [1, \frac{1 + \sqrt{5}}{2}) U (\frac{1 + \sqrt{5}}{2}, \inf)$

I'm not sure how this answer was calculated. Any assistance would be awesome. Thanks.

2. x-sqrt(x+1)!=0

3. Aha! That was a massive oversight.

For other people looking at this thread:

$\displaystyle x - \sqrt{x + 1} = 0$
$\displaystyle x + 1 = x^2$
$\displaystyle x^2 -x - 1 = 0$

$\displaystyle \frac{1 +- \sqrt{1 - 4(1)(-1)}}{2}$
$\displaystyle \frac{1 +- \sqrt{5}}{2}$

Erm, how do I do "+ or -" in Latex so this quadratic eqn makes more sense?

4. but solution with minus is not induced(why?)

5. Because we already know that x must be larger than or equal to 1 to satisfy the numerator.

6. Job well done!

"Erm, how do I do "+ or -" in Latex so this quadratic eqn makes more sense?"

\pm

$\displaystyle \pm$