To find the domain and range of the relation \( R = \{(x, x^3) : x \leq 4, x \in \mathbb{N}\} \), we will follow these steps:
### Step 1: Identify the Domain
The domain of a relation consists of all possible values of \( x \). According to the relation, \( x \) must be a natural number (denoted as \( \mathbb{N} \)) and must also satisfy the condition \( x \leq 4 \).
- The natural numbers less than or equal to 4 are \( 1, 2, 3, \) and \( 4 \).
Thus, the domain of the relation \( R \) is:
\[
\text{Domain} = \{1, 2, 3, 4\}
\]
### Step 2: Calculate the Range
The range of a relation consists of all possible values of \( y \) that correspond to the values of \( x \) in the domain. In this case, \( y \) is defined as \( y = x^3 \).
Now, we will calculate \( y \) for each value of \( x \) in the domain:
- For \( x = 1 \):
\[
y = 1^3 = 1
\]
- For \( x = 2 \):
\[
y = 2^3 = 8
\]
- For \( x = 3 \):
\[
y = 3^3 = 27
\]
- For \( x = 4 \):
\[
y = 4^3 = 64
\]
Now we can compile the range from these calculated values:
\[
\text{Range} = \{1, 8, 27, 64\}
\]
### Final Answer
Thus, the domain and range of the relation \( R \) are:
- Domain: \( \{1, 2, 3, 4\} \)
- Range: \( \{1, 8, 27, 64\} \)
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