To determine the domain and range of the function [tex]\( g(x) = \sqrt{x + 4} \)[/tex], we need to consider the properties of the square root function.
### Step 1: Finding the Domain
The function involves a square root, which is only defined for non-negative values (i.e., values greater than or equal to zero). This means the expression inside the square root, [tex]\( x + 4 \)[/tex], must be non-negative:
[tex]\[ x + 4 \geq 0 \][/tex]
Solving this inequality for [tex]\( x \)[/tex]:
[tex]\[ x \geq -4 \][/tex]
Thus, the domain of [tex]\( g(x) \)[/tex] is all [tex]\( x \)[/tex] such that [tex]\( x \geq -4 \)[/tex]. In interval notation, this can be written as:
[tex]\[ \text{Domain: } [-4, \infty) \][/tex]
### Step 2: Finding the Range
Next, we determine the range of [tex]\( g(x) \)[/tex]. The square root function itself, [tex]\( \sqrt{y} \)[/tex], produces non-negative output values for any non-negative input [tex]\( y \)[/tex].
Since [tex]\( x + 4 \)[/tex] is always non-negative for values of [tex]\( x \)[/tex] in the domain, we examine the possible values of [tex]\( \sqrt{x + 4} \)[/tex]:
- When [tex]\( x = -4 \)[/tex]: [tex]\( g(-4) = \sqrt{-4 + 4} = \sqrt{0} = 0 \)[/tex]
- As [tex]\( x \)[/tex] increases beyond [tex]\(-4\)[/tex], [tex]\( x + 4 \)[/tex] becomes positive, hence [tex]\( \sqrt{x + 4} \)[/tex] increases and can take any value from 0 upwards, without any upper bound.
Therefore, the smallest value of [tex]\( g(x) \)[/tex] is 0 (when [tex]\( x = -4 \)[/tex]), and [tex]\( g(x) \)[/tex] can achieve any value greater than or equal to 0 as [tex]\( x \)[/tex] increases to infinity. So, the range of [tex]\( g(x) \)[/tex] is:
[tex]\[ \text{Range: } [0, \infty) \][/tex]
### Conclusion
Summarizing, we have the domain and range of [tex]\( g(x) = \sqrt{x + 4} \)[/tex]:
- [tex]\( \text{Domain: } [-4, \infty) \)[/tex]
- [tex]\( \text{Range: } [0, \infty) \)[/tex]
From the given options, the correct one is:
[tex]\[ \text{D: } [-4, \infty) \text{ and } R: [0, \infty) \][/tex]
