To find 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 because the square root function is only defined for non-negative values (i.e., the radicand must be greater than or equal to zero).
Here's a step-by-step solution:
1. Identify the radicand:
The radicand is the expression inside the square root, which in this case is [tex]\( x - 5 \)[/tex].
2. Set the radicand greater than or equal to zero:
For the square root function to be defined, we need:
[tex]\[
x - 5 \geq 0
\][/tex]
3. Solve the inequality:
To find the values of [tex]\( x \)[/tex] that satisfy the inequality [tex]\( x - 5 \geq 0 \)[/tex], we add 5 to both sides:
[tex]\[
x \geq 5
\][/tex]
4. Write the domain:
The solution to the inequality [tex]\( x \geq 5 \)[/tex] represents the domain of the function [tex]\( y = \sqrt{x - 5} - 1 \)[/tex]. This means the function is defined for all [tex]\( x \)[/tex] values that are greater than or equal to 5.
Therefore, the domain of [tex]\( y = \sqrt{x - 5} - 1 \)[/tex] is:
[tex]\[
x \geq 5
\][/tex]
