SOLUTION: State the domain of the given function. f(x) = sqrt-x-2 possible answers (-∞, 2] [-2, ∞) (-∞, -2] [2, ∞)

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Question 201546: State the domain of the given function.
f(x) = sqrt-x-2
possible answers
(-∞, 2]
[-2, ∞)
(-∞, -2]
[2, ∞)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
The domain is the set of possible (acceptable) values for x. If a domain is not explicitly specified, then the domain is all Real numbers unless there are reasons to exclude certain numbers. Examples of x-values which should be excluded from domains:
  • x-values that would make a denominator zero
  • x-values that would make the radicand (the number inside a radical) of a square root (or any even-numbered root) negative.
  • x-values that would make the argument of a logarithm function negative


f%28x%29+=+sqrt%28-x-2%29
Your function has no denominators or logarithm functions. But it does have a square root. So we have to make sure that the radicand, (-x-2), can not become negative. In other words we have to make sure the radicand is either 0 or positive. The following inequality says that (-x-2) is either zero or positive:
%28-x-2%29+%3E=+0
To find the domain we need to determine what x-values make this inequality true. In other words we need to solve this inequality. Adding x to both sides we get:
-2+%3E=+x
Note: By adding x, instead of 2, two things are accomplished:
  1. We have the solution in one step!
  2. Perhaps even more importantly, we avoid having to remove the minus in front of x by dividing (or multiplying) both sides by -1. This avoids having to remember to reverse the inequality because we divided (or multiplied) both sides of an inequality by a negative number.
Having the x on the right side does, however, require that we know how to read inequalities correctly. If we read the solution from left to right it says "-2 is greater than or equal to x". If we read it from right to left (Remember, Math is not English. Sometimes should read things "in reverse"), we get "x is less than or equal to -2". Is this a "less than" or a greater than"? The answer: Always read the solution to inequalities starting from where the variable is. So this inequality should be read right to left. In other words: "x is less than or equal to -2".

So this solution describes our domain. In interval notation this would be:
(-∞, -2]