Determine the domain and range of the relation R.
`R={(x+1,x+5)|x in{0,1,2,3,4,5}}`. Draw the graph of R.

Answer

To determine the domain and range of the relation \( R = \{(x+1, x+5) | x \in \{0, 1, 2, 3, 4, 5\}\} \), we will follow these steps:

### Step 1: Identify the values of \( x \)
The values of \( x \) are given as \( \{0, 1, 2, 3, 4, 5\} \).

### Step 2: Calculate the corresponding \( y \) values
Using the relation \( (x+1, x+5) \):
- For \( x = 0 \):
\[ (0+1, 0+5) = (1, 5) \]
- For \( x = 1 \):
\[ (1+1, 1+5) = (2, 6) \]
- For \( x = 2 \):
\[ (2+1, 2+5) = (3, 7) \]
- For \( x = 3 \):
\[ (3+1, 3+5) = (4, 8) \]
- For \( x = 4 \):
\[ (4+1, 4+5) = (5, 9) \]
- For \( x = 5 \):
\[ (5+1, 5+5) = (6, 10) \]

### Step 3: List the ordered pairs
The ordered pairs we have calculated are:
\[ R = \{(1, 5), (2, 6), (3, 7), (4, 8), (5, 9), (6, 10)\} \]

### Step 4: Determine the domain
The domain consists of all the first elements (the \( x \)-values) from the ordered pairs:
\[ \text{Domain} = \{1, 2, 3, 4, 5, 6\} \]

### Step 5: Determine the range
The range consists of all the second elements (the \( y \)-values) from the ordered pairs:
\[ \text{Range} = \{5, 6, 7, 8, 9, 10\} \]

### Step 6: Draw the graph of the relation
To graph the relation, plot the points:
- (1, 5)
- (2, 6)
- (3, 7)
- (4, 8)
- (5, 9)
- (6, 10)

### Final Summary
- **Domain**: \( \{1, 2, 3, 4, 5, 6\} \)
- **Range**: \( \{5, 6, 7, 8, 9, 10\} \)