To determine the domain of the function [tex]\( f(x) = \sqrt{x+4} - 17 \)[/tex], we need to ensure that the expression inside the square root is non-negative. This is important because the square root of a negative number is not defined in the set of real numbers.
Let's break it down step by step:
1. Identify the expression inside the square root:
[tex]\[
\sqrt{x+4}
\][/tex]
2. Set the argument of the square root to be non-negative:
[tex]\[
x + 4 \geq 0
\][/tex]
3. Solve this inequality:
[tex]\[
x + 4 \geq 0
\][/tex]
Subtract 4 from both sides:
[tex]\[
x \geq -4
\][/tex]
Therefore, the domain of the function [tex]\( f(x) = \sqrt{x+4} - 17 \)[/tex] is all [tex]\( x \)[/tex] such that [tex]\( x \geq -4 \)[/tex].
In interval notation, this is expressed as:
[tex]\[
[-4, \infty)
\][/tex]
So, the domain of the function is [tex]\( x \geq -4 \)[/tex].
