If `f: A-> B` and `g: B-> C` be the bijective function, then `(gof)^(-1)` is:

Options

A. `f^(-1)og^(-1)`
B. `fog`
C. `g^(-1)of^(-1)`
D. `gof`

Answer

To solve the problem, we need to find the inverse of the composition of two bijective functions \( g \) and \( f \), denoted as \( (g \circ f)^{-1} \).

### Step-by-Step Solution:

1. **Understand the Composition of Functions**:
The composition \( g \circ f \) means that we first apply the function \( f \) and then apply the function \( g \). So, if \( x \) is in set \( A \), then:
\[ (g \circ f)(x) = g(f(x)) \]
This means \( g \circ f \) is a function from set \( A \) to set \( C \).

2. **Properties of Bijective Functions**:
Since both \( f \) and \( g \) are bijective, they have inverses \( f^{-1} \) and \( g^{-1} \) respectively. A bijective function is both one-to-one (injective) and onto (surjective).

3. **Finding the Inverse of the Composition**:
The inverse of a composition of functions can be expressed as:
\[ (g \circ f)^{-1} = f^{-1} \circ g^{-1} \]
This means that to find the inverse of \( g \circ f \), we first apply \( g^{-1} \) and then \( f^{-1} \).

4. **Determine the Domain and Codomain**:
- The function \( g \circ f \) maps from \( A \) to \( C \).
- Therefore, the inverse \( (g \circ f)^{-1} \) will map from \( C \) to \( A \).

5. **Conclusion**:
Thus, the correct expression for \( (g \circ f)^{-1} \) is:
\[ (g \circ f)^{-1} = f^{-1} \circ g^{-1} \]

### Final Answer:
The correct option is:
\[ (g \circ f)^{-1} = f^{-1} \circ g^{-1} \]