Find the domain and range of the function `f(x)=sqrt(x-5)`.

Answer

To find the domain and range of the function \( f(x) = \sqrt{x - 5} \), we will follow these steps:

### Step 1: Determine the Domain
The function \( f(x) = \sqrt{x - 5} \) involves a square root. For the square root to be defined, the expression inside the square root must be non-negative. Therefore, we need to solve the inequality:

\[ x - 5 \geq 0 \]

### Step 2: Solve the Inequality
To solve the inequality \( x - 5 \geq 0 \), we add 5 to both sides:

\[ x \geq 5 \]

### Step 3: Write the Domain
The domain of the function is all values of \( x \) that satisfy the inequality. Therefore, the domain in interval notation is:

\[ \text{Domain} = [5, \infty) \]

### Step 4: Determine the Range
Next, we will find the range of the function. Since \( f(x) = \sqrt{x - 5} \), we will evaluate the function at the endpoints of the domain.

1. When \( x = 5 \):
\[ f(5) = \sqrt{5 - 5} = \sqrt{0} = 0 \]

2. As \( x \) approaches infinity:
\[ \lim_{x \to \infty} f(x) = \lim_{x \to \infty} \sqrt{x - 5} = \infty \]

### Step 5: Write the Range
From the evaluations, we see that the minimum value of \( f(x) \) is 0 (when \( x = 5 \)) and it increases without bound as \( x \) increases. Therefore, the range of the function in interval notation is:

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

### Final Answer
- **Domain**: \([5, \infty)\)
- **Range**: \([0, \infty)\)

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