To find which intervals are valid domains for a square root function, we need to ensure that the interval starts at or above zero and extends to positive infinity. In mathematical terms, for a function under a square root, [tex]\( f(x) = \sqrt{x} \)[/tex], [tex]\( x \)[/tex] must be non-negative [tex]\( (x \geq 0) \)[/tex].
Now let's match each given interval to see if it fits the requirement for being a valid domain of a square root function:
1. [tex]\((-∞, 6)\)[/tex]: This interval extends from negative infinity to 6. Since it includes negative numbers, it is not suitable for a square root function.
2. [tex]\((1, 4)\)[/tex]: This interval starts from 1 and ends at 4. Both endpoints are positive and the interval does not include any negative numbers. This is a valid domain.
3. [tex]\((-∞, 1)\)[/tex]: This interval extends from negative infinity to 1. Since it includes negative numbers, it is not suitable for a square root function.
4. [tex]\((6, ∞)\)[/tex]: This interval starts from 6 and extends to positive infinity. All values within this interval are non-negative. This is a valid domain.
5. [tex]\((1, ∞)\)[/tex]: This interval starts from 1 and extends to positive infinity. All values within this interval are non-negative. This is a valid domain.
6. [tex]\((3, ∞)\)[/tex]: This interval starts from 3 and extends to positive infinity. All values within this interval are non-negative. This is a valid domain.
7. [tex]\((-2, 6)\)[/tex]: This interval starts from -2 and ends at 6. Since it includes negative numbers, it is not suitable for a square root function.
8. [tex]\((-∞, -2)\)[/tex]: This interval extends from negative infinity to -2. Since it includes negative numbers, it is not suitable for a square root function.
Thus, the intervals that are valid domains for a square root function are:
[tex]\[
(1, 4) \quad (6, \infty) \quad (1, \infty) \quad (3, \infty)
\][/tex]
