If the functions `f` and `g` are given by `f={(1,\ 2),\ (3,\ 5),\ (4,\ 1)}` and `g={(2,\ 3),\ (5,\ 1),\ (1,\ 3)}` , find range of `f` and `g` . Also, write down `fog` and `gof` as sets of ordered pairs.

Explanation

To solve the problem, we need to find the range of the functions \( f \) and \( g \), and then compute the compositions \( f \circ g \) and \( g \circ f \) as sets of ordered pairs.

### Step 1: Identify the functions
The functions are given as:
- \( f = \{(1, 2), (3, 5), (4, 1)\} \)
- \( g = \{(2, 3), (5, 1), (1, 3)\} \)

### Step 2: Find the range of \( f \)
The range of a function consists of all the second elements (outputs) of its ordered pairs.

From the function \( f \):
- The ordered pairs are \( (1, 2), (3, 5), (4, 1) \).
- The outputs are \( 2, 5, 1 \).

Thus, the range of \( f \) is:
\[ \text{Range of } f = \{1, 2, 5\} \]

### Step 3: Find the range of \( g \)
Similarly, we find the range of \( g \).

From the function \( g \):
- The ordered pairs are \( (2, 3), (5, 1), (1, 3) \).
- The outputs are \( 3, 1 \).

Thus, the range of \( g \) is:
\[ \text{Range of } g = \{1, 3\} \]

### Step 4: Compute \( f \circ g \)
To find \( f \circ g \), we need to evaluate \( f(g(x)) \) for each \( x \) in the domain of \( g \).

The domain of \( g \) is \( \{2, 5, 1\} \).

1. For \( g(2) = 3 \):
- Now find \( f(g(2)) = f(3) = 5 \).
- So, \( (2, 5) \) is in \( f \circ g \).

2. For \( g(5) = 1 \):
- Now find \( f(g(5)) = f(1) = 2 \).
- So, \( (5, 2) \) is in \( f \circ g \).

3. For \( g(1) = 3 \):
- Now find \( f(g(1)) = f(3) = 5 \).
- So, \( (1, 5) \) is in \( f \circ g \).

Thus, we have:
\[ f \circ g = \{(2, 5), (5, 2), (1, 5)\} \]

### Step 5: Compute \( g \circ f \)
To find \( g \circ f \), we need to evaluate \( g(f(x)) \) for each \( x \) in the domain of \( f \).

The domain of \( f \) is \( \{1, 3, 4\} \).

1. For \( f(1) = 2 \):
- Now find \( g(f(1)) = g(2) = 3 \).
- So, \( (1, 3) \) is in \( g \circ f \).

2. For \( f(3) = 5 \):
- Now find \( g(f(3)) = g(5) = 1 \).
- So, \( (3, 1) \) is in \( g \circ f \).

3. For \( f(4) = 1 \):
- Now find \( g(f(4)) = g(1) = 3 \).
- So, \( (4, 3) \) is in \( g \circ f \).

Thus, we have:
\[ g \circ f = \{(1, 3), (3, 1), (4, 3)\} \]

### Final Results
- Range of \( f \): \( \{1, 2, 5\} \)
- Range of \( g \): \( \{1, 3\} \)
- \( f \circ g = \{(2, 5), (5, 2), (1, 5)\} \)
- \( g \circ f = \{(1, 3), (3, 1), (4, 3)\} \)