Find the domain and the range of the real function f defined by `f(x)=sqrt((x-1))`.

Explanation

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

### Step 1: Determine the Domain
The function \( f(x) = \sqrt{x - 1} \) involves a square root. The expression inside the square root must be non-negative for the function to be defined. Therefore, we need to solve the inequality:

\[ x - 1 \geq 0 \]

### Step 2: Solve the Inequality
To solve the inequality \( x - 1 \geq 0 \), we add 1 to both sides:

\[ x \geq 1 \]

### Step 3: Write the Domain
The domain of the function is all values of \( x \) that satisfy the inequality \( x \geq 1 \). In interval notation, this is expressed as:

\[ \text{Domain} = [1, \infty) \]

### Step 4: Determine the Range
Next, we will find the range of the function. Since \( f(x) = \sqrt{x - 1} \) and \( x \) must be at least 1 (from the domain), we can find the minimum value of \( f(x) \):

- When \( x = 1 \):
\[ f(1) = \sqrt{1 - 1} = \sqrt{0} = 0 \]

As \( x \) increases beyond 1, \( f(x) \) will also increase without bound. Therefore, the smallest value of \( f(x) \) is 0, and there is no upper limit.

### Step 5: Write the Range
Thus, the range of the function is:

\[ \text{Range} = [0, \infty) \]

### Final Answer
- **Domain**: \([1, \infty)\)
- **Range**: \([0, \infty)\)

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