Let's analyze the function [tex]\( g(x) = \sqrt{x + 3} \)[/tex] to determine its domain and range.
### Domain
The domain of a function is the set of all possible input values (x-values) that will produce a valid output (y-value). For the function [tex]\( g(x) = \sqrt{x + 3} \)[/tex], the expression inside the square root, [tex]\( x + 3 \)[/tex], must be non-negative because the square root of a negative number is not defined in the real number system. Therefore, we require:
[tex]\[ x + 3 \geq 0 \][/tex]
Solving this inequality for [tex]\( x \)[/tex],
[tex]\[ x \geq -3 \][/tex]
So, the domain is:
[tex]\[ D: [-3, \infty) \][/tex]
### Range
The range of a function is the set of all possible output values (y-values) it can produce. For [tex]\( g(x) = \sqrt{x + 3} \)[/tex], the square root function only produces non-negative values, starting from 0 and increasing as [tex]\( x \)[/tex] increases. Therefore, the range is:
[tex]\[ R: [0, \infty) \][/tex]
### Multiple Choice Answer
Given the given choices:
1. [tex]\( D: [3, \infty) \)[/tex] and [tex]\( R: [0, \infty) \)[/tex]
2. [tex]\( D: [-3, \infty) \)[/tex] and [tex]\( R: [0, \infty) \)[/tex]
3. [tex]\( D: (-3, \infty) \)[/tex] and [tex]\( R: (-\infty, 0) \)[/tex]
The correct domain and range for [tex]\( g(x) = \sqrt{x + 3} \)[/tex] are:
[tex]\[ D: [-3, \infty) \][/tex]
[tex]\[ R: [0, \infty) \][/tex]
Therefore, the correct answer is the second choice:
[tex]\[ \boxed{D: [-3, \infty) \text{ and } R: [0, \infty)} \][/tex]
