Question

Find the domain of the given function. f(x) = square root of x +3 / ( x+8 ) ( x -2 )

Answer 1

is this what you mean? \[\sqrt{\frac{ x+3 }{ (x+8)(x-2) }}\]

Answer 2

if so, then we need to make sure the radicand is > or = 0, and that the denominator is nonzero.
So for the denominator: need \[x \neq-8\] and \[x \neq2\].
Now for the radicand, set up a sign graph of the factors. The partition numbers are x=-8, x=-3, x=2.
For \[(-\infty,-8)\] we have (negative)/[negative x negative] so the radicand will be negative. So this is not in the domain.
For (-8,-3) we have (negative)/[positive x negative] so the radicand will be positive. So this IS in the domain.
At x=-3, the radicand=0, so this is in the domain.
For (-3,2), we have (positive)/[positive x negative] so the radicand will be negative. This is NOT in the domain.
For x>2, all is positive so radicand is positive and this is in the domain.
So the complete domain is: \[(-8,-3]\cup(2,\infty)\]

Answer 3

The square root is on the numerator only

Answer 4

OK, then all above applies except no need to worry about the sign of the denominator, only that it not =0. Need x+3> or =0 so that the radicand is non-negative, so that is \[[-3,\infty)\] but also need x not =2. So the domain is:\[[-3,2)\cup(2,\infty)\]

Answer 5

So it would be x ≥ -3, x ≠ 2

Answer 6

Yes.