To determine the domain of the given radical function [tex]\(f(x) = \sqrt{x + 1} - 3\)[/tex], we need to ensure that the expression inside the square root is non-negative because the square root of a negative number is not defined in the set of real numbers.
The expression inside the square root is [tex]\(x + 1\)[/tex]. We need to solve for when this expression is non-negative:
[tex]\[ x + 1 \geq 0 \][/tex]
Subtract 1 from both sides to isolate [tex]\(x\)[/tex]:
[tex]\[ x \geq -1 \][/tex]
Hence, the domain of the function [tex]\(f(x) = \sqrt{x + 1} - 3\)[/tex] includes all [tex]\(x\)[/tex] values that are greater than or equal to [tex]\(-1\)[/tex]. We can express this domain in interval notation as:
[tex]\[ [-1, \infty) \][/tex]
Therefore, the correct answer is:
[tex]\[
\text{D. } [-1, \infty)
\][/tex]
