Find the domain the range of each of the following function: `f(x)=x!`

Explanation

To find the domain and range of the function \( f(x) = x! \) (x factorial), we will analyze the function step by step.

### Step 1: Determine the Domain
The factorial function \( x! \) is defined only for non-negative integers. This means:
- \( x \) can be 0, 1, 2, 3, and so on.
- Factorial is not defined for negative integers or non-integer values.

Thus, the domain of \( f(x) = x! \) is the set of whole numbers (non-negative integers).

**Domain:**
\[ \text{Domain} = \{0, 1, 2, 3, \ldots\} = \mathbb{N}_0 \]

### Step 2: Determine the Range
Next, we need to find the range of the function. The factorial of a non-negative integer \( n \) is defined as:
- \( 0! = 1 \)
- \( 1! = 1 \)
- \( 2! = 2 \)
- \( 3! = 6 \)
- \( 4! = 24 \)
- \( 5! = 120 \)
- and so on.

As \( n \) increases, \( n! \) increases rapidly. The values of \( n! \) for \( n = 0, 1, 2, 3, \ldots \) are:
- \( f(0) = 1 \)
- \( f(1) = 1 \)
- \( f(2) = 2 \)
- \( f(3) = 6 \)
- \( f(4) = 24 \)
- \( f(5) = 120 \)

From this, we can see that the range consists of all the values of \( n! \) where \( n \) is a non-negative integer. The values start from 1 and increase without bound.

**Range:**
\[ \text{Range} = \{ n! \mid n \in \mathbb{N}_0 \} = \{ 1, 1, 2, 6, 24, 120, \ldots \} \]

### Final Result
- **Domain:** \( \{0, 1, 2, 3, \ldots\} \)
- **Range:** \( \{ n! \mid n \in \mathbb{N}_0 \} \)