To determine the domain and range for the function [tex]\( g(x) = \sqrt{x + 3} \)[/tex], we need to consider the following aspects:
### Domain
The domain of a function includes all the values of [tex]\( x \)[/tex] for which the function is defined. Since [tex]\( g(x) \)[/tex] involves a square root, we need to ensure the argument inside the square root is non-negative (because the square root of a negative number is not a real number).
1. The expression inside the square root is [tex]\( x + 3 \)[/tex].
2. Set the expression inside the square root [tex]\(\geq 0\)[/tex]:
[tex]\[
x + 3 \geq 0
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[
x \geq -3
\][/tex]
Thus, the domain of [tex]\( g(x) = \sqrt{x + 3} \)[/tex] is all [tex]\( x \)[/tex] such that [tex]\( x \geq -3 \)[/tex].
### Range
The range of a function includes all the possible output values (or [tex]\( g(x) \)[/tex]).
1. The output [tex]\( g(x) = \sqrt{x + 3} \)[/tex] will always yield non-negative values since the square root of any non-negative number is also non-negative.
2. The smallest value inside the square root is obtained when [tex]\( x = -3 \)[/tex]:
[tex]\[
g(-3) = \sqrt{-3 + 3} = \sqrt{0} = 0
\][/tex]
3. As [tex]\( x \)[/tex] increases from [tex]\(-3\)[/tex] to [tex]\(\infty\)[/tex], the output also increases from [tex]\( 0 \)[/tex] to [tex]\(\infty \)[/tex].
Thus, the range of [tex]\( g(x) = \sqrt{x + 3} \)[/tex] is all non-negative numbers, starting from [tex]\( 0 \)[/tex] and increasing without bound.
### Conclusion
- The domain of [tex]\( g(x) = \sqrt{x + 3} \)[/tex] is [tex]\( [-3, \infty) \)[/tex].
- The range of [tex]\( g(x) = \sqrt{x + 3} \)[/tex] is [tex]\( [0, \infty) \)[/tex].
So, the correct answer is:
[tex]\[ \text{D: } [-3, \infty) \text{ and R: } [0, \infty) \][/tex]
