Let N be the set of natural numbers. Describe the following relations in words, giving their domain and the range. `{(1,4),(5,16),(7,22),(12,37)}`

Answer

To describe the relation given by the set of ordered pairs \(\{(1,4),(5,16),(7,22),(12,37)\}\), we will follow these steps:

### Step 1: Identify the Ordered Pairs
The relation consists of the following ordered pairs:
- (1, 4)
- (5, 16)
- (7, 22)
- (12, 37)

### Step 2: Define Domain and Range
- **Domain**: The domain of a relation is the set of all first elements (inputs) of the ordered pairs.
- **Range**: The range of a relation is the set of all second elements (outputs) of the ordered pairs.

### Step 3: Extract the Domain
From the ordered pairs, we can extract the first elements:
- The first elements are: 1, 5, 7, and 12.
- Thus, the domain of the relation is:
\[ \text{Domain} = \{1, 5, 7, 12\} \]

### Step 4: Extract the Range
From the ordered pairs, we can extract the second elements:
- The second elements are: 4, 16, 22, and 37.
- Thus, the range of the relation is:
\[ \text{Range} = \{4, 16, 22, 37\} \]

### Step 5: Describe the Relation in Words
The relation can be described as follows:
- For each natural number \(x\) in the domain, the corresponding value \(y\) in the range is obtained by the formula \(y = 3x + 1\).
- Specifically, when \(x = 1\), \(y = 4\); when \(x = 5\), \(y = 16\); when \(x = 7\), \(y = 22\); and when \(x = 12\), \(y = 37\).

### Final Summary
- **Domain**: \(\{1, 5, 7, 12\}\)
- **Range**: \(\{4, 16, 22, 37\}\)

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