Let's determine the domain of the function [tex]\( f(x) = \sqrt{x-1} \)[/tex].
### Step-by-Step Solution:
1. Understand the Function:
- The function [tex]\( f(x) = \sqrt{x-1} \)[/tex] involves a square root. For the square root to be defined in the real numbers, the expression inside the square root must be non-negative.
2. Set Up the Inequality:
- Since the square root function is only defined for non-negative values, we need [tex]\( x-1 \geq 0 \)[/tex].
3. Solve the Inequality:
- Solving the inequality [tex]\( x-1 \geq 0 \)[/tex]:
[tex]\[
x-1 \geq 0 \implies x \geq 1
\][/tex]
4. Determine the Domain:
- The solution [tex]\( x \geq 1 \)[/tex] implies that [tex]\( x \)[/tex] can be 1 or any number greater than 1. Therefore, [tex]\( x \)[/tex] ranges from 1 to infinity, including 1 itself.
5. Interval Notation:
- In interval notation, this is written as [tex]\( [1, \infty) \)[/tex].
So the domain of [tex]\( f(x) = \sqrt{x-1} \)[/tex] is [tex]\( [1, \infty) \)[/tex].
### Conclusion:
The correct option for the domain of [tex]\( f \)[/tex] is:
(b) [tex]\( [1, \infty) \)[/tex]
