Find the domain and the range of the relation R given by R = { (x,y) : `y = x+6/x` , where x ` y in N and x lt 6 `}

Answer

To find the domain and range of the relation \( R \) given by \( R = \{ (x,y) : y = x + \frac{6}{x} \} \), where \( x, y \in \mathbb{N} \) (natural numbers) and \( x < 6 \), we will follow these steps:

### Step 1: Identify the Domain
The domain is defined by the values of \( x \) that satisfy the given conditions. Since \( x \) must be a natural number and less than 6, we can list the possible values:

- Natural numbers less than 6 are: \( 1, 2, 3, 4, 5 \).

Thus, the domain of \( R \) is:
\[ \text{Domain} = \{ 1, 2, 3, 4, 5 \} \]

### Step 2: Calculate Corresponding Values of \( y \)
Next, we will calculate the corresponding values of \( y \) for each value of \( x \) in the domain using the equation \( y = x + \frac{6}{x} \).

1. For \( x = 1 \):
\[ y = 1 + \frac{6}{1} = 1 + 6 = 7 \]

2. For \( x = 2 \):
\[ y = 2 + \frac{6}{2} = 2 + 3 = 5 \]

3. For \( x = 3 \):
\[ y = 3 + \frac{6}{3} = 3 + 2 = 5 \]

4. For \( x = 4 \):
\[ y = 4 + \frac{6}{4} = 4 + 1.5 = 5.5 \]

5. For \( x = 5 \):
\[ y = 5 + \frac{6}{5} = 5 + 1.2 = 6.2 \]

### Step 3: Identify Valid Values of \( y \)
Since both \( x \) and \( y \) must be natural numbers, we will filter the calculated values of \( y \):

- For \( x = 1 \), \( y = 7 \) (valid)
- For \( x = 2 \), \( y = 5 \) (valid)
- For \( x = 3 \), \( y = 5 \) (valid)
- For \( x = 4 \), \( y = 5.5 \) (not valid)
- For \( x = 5 \), \( y = 6.2 \) (not valid)

Thus, the valid values of \( y \) are \( 5 \) and \( 7 \).

### Step 4: Determine the Range
The range of the relation \( R \) is the set of valid \( y \) values:

\[ \text{Range} = \{ 5, 7 \} \]

### Final Answer
- **Domain**: \( \{ 1, 2, 3, 4, 5 \} \)
- **Range**: \( \{ 5, 7 \} \)