Maesyn:
When you deal with SQRT function you have to make sure that you are taking the square-root of a non-negative number (> or = 0). Ask yourself, for what values of x (3x+5)/(x-4) is non-negative.
First of all, you can not have x = 4, because if x = 4, x - 4 = 0 and you can not divide by zero to get any meaningful answer.
Further, for (3x+5)/(x-4) to be non-negative, (a) the numerator is non-negative, AND the denominator must be negative at the same time (remember, the latter can not be zero, from above). For (a), (3x + 5) < or = 0 means 3x < or = - 5, that is x < or = - 5/3. For (b), (x-4) < 0 means x < 4. For (a) and (b) to hold at the same time, x < or = - 5/3.
Now, for (3x+5)/(x-4) to be non-negative, (c) if the numerator is non-negative, (d) the denominator must be positive. For (c) 3x + 5 > or = 0 means 3x + 5 > or = 0, that is, x > or = -5/3. For (d) to hold, x- 4 > 0 means x > 4. For (c) AND (d) to hold, we must have x > 4
Combine the inequalities in bold letters. If x < or = -5/3 OR if x > 4, then (3x + 5) / (x - 4) is non-negative.
The above domain can be written as {x : either x < or = -5/3 or x > 4}, to be read "the set of all (real numbers) x such that either x is at most (-5/3) or greater than 4."
Maesyn, it is always good to take some test values, say x= -2, or, x = 5, and some more, plug into (3x + 5)/(x - 4) and check that it is non-negative so that you can take the square-root.
Dattaprabhakar (Dr. G.)
