gof exists, when :

Options

A. Domain of f = domain of g
B. Co-domain of f = domain of g
C. Co-domain of g = domain of g
D. Co- domain of g = co-domain of f

Answer

To determine when the composition of two functions \( g \) and \( f \) (denoted as \( g \circ f \)) exists, we need to analyze the relationship between the range of \( f \) and the domain of \( g \).

### Step-by-Step Solution:

1. **Understanding Function Composition**:
The composition of two functions \( g \) and \( f \) is defined as \( g(f(x)) \). For this composition to be valid, the output of function \( f \) must be a valid input for function \( g \).

2. **Identifying the Range of \( f \)**:
The range of function \( f \) is the set of all possible output values that \( f \) can produce. This is denoted as \( \text{Range}(f) \).

3. **Identifying the Domain of \( g \)**:
The domain of function \( g \) is the set of all possible input values that \( g \) can accept. This is denoted as \( \text{Domain}(g) \).

4. **Condition for Composition**:
For the composition \( g(f(x)) \) to exist for all \( x \) in the domain of \( f \), the range of \( f \) must be a subset of the domain of \( g \). Mathematically, this can be expressed as:
\[ \text{Range}(f) \subseteq \text{Domain}(g) \]

5. **Conclusion**:
Therefore, the composition \( g \circ f \) exists if and only if the range of \( f \) is a subset of the domain of \( g \).

### Final Answer:
The composition \( g \circ f \) exists when the range of \( f \) is a subset of the domain of \( g \).

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