To determine the domain of the function [tex]\( y = \sqrt{x - 5} - 1 \)[/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.
Here are the steps to find the domain:
1. Identify the expression inside the square root:
[tex]\[
x - 5
\][/tex]
2. Set up the inequality for the expression to be non-negative:
[tex]\[
x - 5 \geq 0
\][/tex]
3. Solve the inequality:
[tex]\[
x \geq 5
\][/tex]
This means that the variable [tex]\( x \)[/tex] must be greater than or equal to 5 for the function [tex]\( y = \sqrt{x - 5} - 1 \)[/tex] to be defined.
Hence, the domain of the function [tex]\( y = \sqrt{x - 5} - 1 \)[/tex] is:
[tex]\[
[5, \infty)
\][/tex]
This indicates that the function is defined for all [tex]\( x \)[/tex] values starting from 5 and continuing to positive infinity.
