Determine the domain and range of the relation R.
`R={(x,x^(3))|x" is a prime number less than 10"}`

Answer

To determine the domain and range of the relation \( R = \{(x, x^3) | x \text{ is a prime number less than 10}\} \), we will follow these steps:

### Step 1: Identify the prime numbers less than 10.
The prime numbers less than 10 are:
- 2
- 3
- 5
- 7

### Step 2: Define the domain.
The domain of the relation \( R \) consists of the values of \( x \) that we identified in Step 1. Therefore, the domain is:
\[ \text{Domain} = \{2, 3, 5, 7\} \]

### Step 3: Calculate the corresponding \( y \) values using the function \( y = x^3 \).
Now we will calculate \( y \) for each \( x \) in the domain:
- For \( x = 2 \):
\[ y = 2^3 = 8 \]
- For \( x = 3 \):
\[ y = 3^3 = 27 \]
- For \( x = 5 \):
\[ y = 5^3 = 125 \]
- For \( x = 7 \):
\[ y = 7^3 = 343 \]

### Step 4: Define the range.
The range of the relation \( R \) consists of the calculated \( y \) values. Therefore, the range is:
\[ \text{Range} = \{8, 27, 125, 343\} \]

### Final Result:
- **Domain**: \( \{2, 3, 5, 7\} \)
- **Range**: \( \{8, 27, 125, 343\} \)

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