To determine the domain of the function [tex]\( f(x) = \sqrt{x+5} - 1 \)[/tex], we need to ensure that the expression inside the square root is non-negative. This is because the square root function is only defined for non-negative numbers.
Let's analyze the expression [tex]\( x + 5 \)[/tex]:
1. Non-Negativity Condition:
[tex]\[
x + 5 \geq 0
\][/tex]
2. Solving for [tex]\( x \)[/tex]:
[tex]\[
x + 5 \geq 0 \\
x \geq -5
\][/tex]
3. Interpreting the Result:
- The inequality [tex]\( x \geq -5 \)[/tex] means that [tex]\( x \)[/tex] can be any number greater than or equal to [tex]\(-5\)[/tex].
4. Domain in Interval Notation:
- The domain includes all [tex]\( x \)[/tex] values that are greater than or equal to [tex]\(-5\)[/tex].
- In interval notation, this is written as:
[tex]\[
[-5, \infty)
\][/tex]
Therefore, the correct domain of the function [tex]\( f(x) = \sqrt{x+5} - 1 \)[/tex] written in interval notation is:
[tex]\[ \boxed{[-5, \infty)} \][/tex]
