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i am assuming that x-2 is completely under the radical (that's why i hate this radical symbol, it's subjective).

sqrt(9)=3, sqrt(25)=5, sqrt(-81)=getting the math police called on you

you can not have a negative under the radical (because we only want a real number range, sqrts of negative numbers are imaginary)

so taking that into account, we know that x values that create a negative number under the radical must be excluded from the domain. clearly numbers less than two will cause that. However here is a mathematical reason for that. 0 is the lowest number that is not negative. It is one of the properties of real numbers that if a number is greater than (or equal to) 0, then the square will also be greater than (or equal to) zero. Also note that sqrt(0)=0 which is acceptable. let =< mean less than or equal to

0=<sqrt(x-2) implies that 0^2=<sqrt(x-2)^2 which means 0=<x-2. by adding 2 on both sides, you can see that 2=<x.

even though this is equivalent to [2,inf), it's clear your teacher wants you to practice interval notation. so here are the mechanics

[ and ( are lower bounds, meaning the lowest number in the interval

] and ) are upper bounds, meaning the highest number in the interval

a U like symbol is used to mean union when there is more than one acceptable interval

[ and ] are used when the lower or upper bound CAN be used in the domain. so in this case [2 means 2 is the lowest number and it CAN be used (because sqrt(2-2)=sqrt(0)=0 which is fine)

( and ) are used when this is the lower or upper bound cannot work. the inequality 5<x<7 would be (5,7) because 5 and 7 are the bounds, but they cannot satisfy the inequality. -inf and inf must ALWAYS be used with ( or ) respectively because no number can actually equal +/-inf

enjoy :)

In an interval, the first number to the left of the comma is the left-most number in the range of x to y (x to y being two different numbers, x being less than y and in this case x being 2). Think of the number line when you see this:

[0, 9]

This means all the numbers from 0 to 9. The brackets mean "inclusive", so you will have 0 in the set of 0 to 9, and have 9 in the set from 0 to 9. The parenthesis mean exclusive. You omit the x and y in [x, y] from the set of numbers from x to y. Take this example:

(4, 9)

This means {5, 6, 7, 8}. Notice how we keep the 4 and 9 out.

You can have both the parenthesis and brackets in interval notation:

(4, 9]

This means {5, 6, 7, 8, 9}.