Find the domain and the range of the real function f defined by `f(x)=sqrt((x-1))`.(a)Domain = ( 1 , ∞ ) , Range = ( 0 , ∞ ) (b) Domain = [ 1 , ∞ ) , Range = ( 0 , ∞ (c) Domain = ( 1 , ∞ ) , Range = [ 0 , ∞ ) (d)Domain = [ 1 , ∞ ) , Range = [ 0 , ∞ )

Options

A. Domain`=(1,oo),` Range `= (0,oo)`
B. Domain`=[1,oo),` Range `= (0,oo)`
C. Domain`=(1,oo),` Range `= [0,oo)`
D. Domain`=[1,oo),` Range `= [0,oo)`

Explanation

To find the domain and range of the function \( f(x) = \sqrt{x - 1} \), we will follow the steps below:

### Step 1: Determine the Domain
The domain of a function is the set of all possible input values (x-values) for which the function is defined.

1. The expression inside the square root must be non-negative because the square root of a negative number is not defined in the real number system. Therefore, we set up the inequality:
\[ x - 1 \geq 0 \]
2. Solving this inequality gives:
\[ x \geq 1 \]
3. Thus, the domain of the function is all real numbers \( x \) such that \( x \) is greater than or equal to 1. In interval notation, this is expressed as:
\[ \text{Domain} = [1, \infty) \]

### Step 2: Determine the Range
The range of a function is the set of all possible output values (f(x)-values).

1. Since \( f(x) = \sqrt{x - 1} \), we need to find the minimum value of \( f(x) \) when \( x \) is at its minimum (which is 1).
2. When \( x = 1 \):
\[ f(1) = \sqrt{1 - 1} = \sqrt{0} = 0 \]
3. As \( x \) increases beyond 1, \( f(x) \) will also increase without bound. Therefore, the function can take all values from 0 to infinity.
4. Thus, the range of the function is:
\[ \text{Range} = [0, \infty) \]

### Final Answer
Combining both results, we have:
- Domain: \([1, \infty)\)
- Range: \([0, \infty)\)

The correct option is (d) Domain = \([1, \infty)\), Range = \([0, \infty)\).