Express the following functions in the form of order pairs and find their range:
(i) `f:A to R, f(x)=x^(2)+5`, where `A={-2,-1,0,1}`
`g:A to N, g(x)=3x`, where `A:{x:x in N, x lt 4}`

Answer

To solve the given problem, we will express the functions in the form of ordered pairs and then find their range step by step.

### Part (i): Function \( f: A \to \mathbb{R}, f(x) = x^2 + 5 \)

**Step 1: Identify the set \( A \)**
The set \( A \) is given as \( A = \{-2, -1, 0, 1\} \).

**Step 2: Calculate \( f(x) \) for each element in \( A \)**
We will compute \( f(x) \) for each value in \( A \):

- For \( x = -2 \):
\[ f(-2) = (-2)^2 + 5 = 4 + 5 = 9 \]

- For \( x = -1 \):
\[ f(-1) = (-1)^2 + 5 = 1 + 5 = 6 \]

- For \( x = 0 \):
\[ f(0) = 0^2 + 5 = 0 + 5 = 5 \]

- For \( x = 1 \):
\[ f(1) = 1^2 + 5 = 1 + 5 = 6 \]

**Step 3: Write the ordered pairs**
Now we can express the function \( f \) in the form of ordered pairs:
\[ f = \{(-2, 9), (-1, 6), (0, 5), (1, 6)\} \]

**Step 4: Find the range of \( f \)**
The range is the set of all output values from the function:
\[ \text{Range of } f = \{9, 6, 5\} \]

### Part (ii): Function \( g: A \to \mathbb{N}, g(x) = 3x \)

**Step 1: Identify the set \( A \)**
The set \( A \) is defined as \( A = \{x : x \in \mathbb{N}, x < 4\} \). The natural numbers less than 4 are \( 1, 2, 3 \).

**Step 2: Calculate \( g(x) \) for each element in \( A \)**
We will compute \( g(x) \) for each value in \( A \):

- For \( x = 1 \):
\[ g(1) = 3 \times 1 = 3 \]

- For \( x = 2 \):
\[ g(2) = 3 \times 2 = 6 \]

- For \( x = 3 \):
\[ g(3) = 3 \times 3 = 9 \]

**Step 3: Write the ordered pairs**
Now we can express the function \( g \) in the form of ordered pairs:
\[ g = \{(1, 3), (2, 6), (3, 9)\} \]

**Step 4: Find the range of \( g \)**
The range is the set of all output values from the function:
\[ \text{Range of } g = \{3, 6, 9\} \]

### Summary of Results
- For the function \( f \):
- Ordered pairs: \( \{(-2, 9), (-1, 6), (0, 5), (1, 6)\} \)
- Range: \( \{9, 6, 5\} \)

- For the function \( g \):
- Ordered pairs: \( \{(1, 3), (2, 6), (3, 9)\} \)
- Range: \( \{3, 6, 9\} \)