To determine the domain of the function [tex]\( y = \sqrt{x - 5} - 1 \)[/tex], we have to identify all the possible values of [tex]\( x \)[/tex] for which the expression is defined.
The primary concern here is the square root function [tex]\( \sqrt{x - 5} \)[/tex]. The square root function is only defined for non-negative values; that is, the expression inside the square root must be greater than or equal to zero.
Let's set up the inequality:
[tex]\[ x - 5 \geq 0 \][/tex]
To find the valid values of [tex]\( x \)[/tex], solve this inequality:
[tex]\[ x \geq 5 \][/tex]
This tells us that [tex]\( x \)[/tex] must be greater than or equal to 5 for the function [tex]\( \sqrt{x - 5} - 1 \)[/tex] to be defined. Therefore, for all [tex]\( x \geq 5 \)[/tex], the value inside the square root is non-negative, and the function will be defined.
Hence, the domain of the function [tex]\( y = \sqrt{x - 5} - 1 \)[/tex] is:
[tex]\[ [5, \infty) \][/tex]
This means that all real numbers [tex]\( x \)[/tex] starting from 5 and onwards (including 5) are in the domain of the function [tex]\( y = \sqrt{x - 5} - 1 \)[/tex].
