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 domain of a function is the set of all possible input values (x-values) for which the function is defined.

For the function \( f(x) = \sqrt{x - 5} \), the expression inside the square root must be non-negative (greater than or equal to zero) because the square root of a negative number is not defined in the real number system.

Set up the inequality:
\[ x - 5 \geq 0 \]

### Step 2: Solve the Inequality
To solve the inequality, we add 5 to both sides:
\[ x \geq 5 \]

### Step 3: Write the Domain
The domain of the function is all x-values that satisfy the inequality. Therefore, we can express the domain in interval notation:
\[ \text{Domain} = [5, \infty) \]

### Step 4: Determine the Range
Next, we find the range of the function. The range is the set of all possible output values (y-values) of the function.

Since \( f(x) = \sqrt{x - 5} \), we can analyze the output:
- When \( x = 5 \), \( f(5) = \sqrt{5 - 5} = \sqrt{0} = 0 \).
- As \( x \) increases beyond 5, \( f(x) \) will also increase because the square root function is increasing.

Thus, as \( x \) approaches infinity, \( f(x) \) also approaches infinity:
\[ f(x) \to \infty \text{ as } x \to \infty \]

### Step 5: Write the Range
The minimum value of \( f(x) \) is 0 (when \( x = 5 \)), and there is no upper limit. Therefore, the range of the function can be expressed in interval notation:
\[ \text{Range} = [0, \infty) \]

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