Find the domain and range of each of the following functions
`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 expression under the square root must be non-negative for the function to be defined. Therefore, we set up the inequality:

\[ x - 5 \geq 0 \]

Solving this inequality:

\[ x \geq 5 \]

This means that the function is defined for all \( x \) values greater than or equal to 5. Thus, the domain of the function is:

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

### Step 2: Determine the Range
Next, we need to find the range of the function. We start by expressing \( f(x) \) in terms of \( y \):

\[ y = \sqrt{x - 5} \]

To find \( x \) in terms of \( y \), we square both sides:

\[ y^2 = x - 5 \]

Rearranging gives:

\[ x = y^2 + 5 \]

Since \( x \) must be greater than or equal to 5 (from our domain), we can substitute this back into our equation:

\[ y^2 + 5 \geq 5 \]

This simplifies to:

\[ y^2 \geq 0 \]

Since \( y^2 \) is always non-negative, the smallest value \( y \) can take is 0. As \( x \) increases beyond 5, \( y \) will also increase. Therefore, the range of the function is:

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

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