If `f:{1,3} rarr {1,2,5} "and " g:{1,2,5} rarr{1,2,3,4}` be given by `f={(1,2),(3,5)}, g={(1,3),(2,3),(5,1)}`,
write g of.

Answer

To find the composition of functions \( g \) and \( f \) (denoted as \( g \circ f \)), we will follow these steps:

### Step 1: Identify the functions
We have two functions:
- \( f: \{1, 3\} \to \{1, 2, 5\} \) defined by \( f = \{(1, 2), (3, 5)\} \)
- \( g: \{1, 2, 5\} \to \{1, 2, 3, 4\} \) defined by \( g = \{(1, 3), (2, 3), (5, 1)\} \)

### Step 2: Find \( g(f(x)) \) for each \( x \) in the domain of \( f \)
We will evaluate \( g(f(1)) \) and \( g(f(3)) \).

#### Step 2.1: Calculate \( f(1) \)
From the function \( f \):
- \( f(1) = 2 \)

#### Step 2.2: Calculate \( g(f(1)) = g(2) \)
From the function \( g \):
- \( g(2) = 3 \)

So, we have \( g(f(1)) = 3 \).

#### Step 2.3: Calculate \( f(3) \)
From the function \( f \):
- \( f(3) = 5 \)

#### Step 2.4: Calculate \( g(f(3)) = g(5) \)
From the function \( g \):
- \( g(5) = 1 \)

So, we have \( g(f(3)) = 1 \).

### Step 3: Compile the results
Now we can compile the results of \( g(f(x)) \):
- For \( x = 1 \), \( g(f(1)) = 3 \)
- For \( x = 3 \), \( g(f(3)) = 1 \)

Thus, the composition \( g \circ f \) can be represented as:
\[ g \circ f = \{(1, 3), (3, 1)\} \]

### Final Answer
The composition \( g \circ f \) is \( \{(1, 3), (3, 1)\} \).

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