If `f:AtoB` and `g:BtoC` are both bijective functions then `(gof)^(-1)` is

Options

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

Answer

To solve the problem, we need to find the inverse of the composition of two bijective functions \( g \) and \( f \). Let's denote the functions as follows:

- \( f: A \to B \) (a bijective function)
- \( g: B \to C \) (a bijective function)

We want to find \( (g \circ f)^{-1} \).

### Step-by-Step Solution:

1. **Understanding the Composition of Functions**:
The composition of two functions \( g \) and \( f \) is defined as:
\[ (g \circ f)(x) = g(f(x)) \]
for all \( x \in A \).

2. **Using the Property of Inverses**:
For any two bijective functions \( f \) and \( g \), the inverse of their composition can be expressed as:
\[ (g \circ f)^{-1} = f^{-1} \circ g^{-1} \]
This property holds because applying the inverse functions in reverse order will yield the original input.

3. **Verifying the Inverse**:
To verify this, we can check:
\[ (g \circ f)(f^{-1}(y)) = g(f(f^{-1}(y))) = g(y) \]
and
\[ (f^{-1} \circ g^{-1})(y) = f^{-1}(g^{-1}(y)) \]
Both expressions should yield \( y \) if our inverse is correct.

4. **Conclusion**:
Since both expressions yield the same result, we conclude that:
\[ (g \circ f)^{-1} = f^{-1} \circ g^{-1} \]

### Final Answer:
Thus, the inverse of the composition \( (g \circ f)^{-1} \) is:
\[ (g \circ f)^{-1} = f^{-1} \circ g^{-1} \]