Thread: Quick question

find and sketch the domain of f(x,y) = sqrt(1+ln(xy))
it will be:
y>1/(ex)
graph of y=1/ex is approximately the same as y=1/x
where should i shade it ??
it have curve in quadratic 1 and curve in quadratic 3
i will shade it inside both curves or just the curve in the quadratic 1??

i think,
since y > 1/ex
---> xy > 1/e
product is positive
so I will shade the curve in quadratic 3
right??!

If our function is



then is defined exactly where is defined and non-negative; that is, where



By two reversible operations, we find that this is equivalent to saying



Therefore, must be positive and must lie in the first or third quadrant. Now, if we knew that were positive, we could divide by to obtain



but in the third quadrant, , and we can't do this without reversing the inequality. In fact, while your answer is correct in the first quadrant, in the third quadrant it will be



Hope this helps!