Statement-1 : `fog = gof rArr f^(-1) = g` or `g^(-1) = f`.
and
Statement-2 : `fog != gof`.

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is True, Statement-2 is NOT acorrect explanation for Statement-1.
C. Statement -1 is False, Statement -2 is False
D. Statement -1 is False, Statement -2 is True

Answer

To analyze the two statements given in the question, we will break down the concepts of function composition and their properties step by step.

### Step-by-Step Solution

1. **Understanding Functions and Composition**:
Let \( f: A \to B \) and \( g: B \to C \) be two functions. The composition of these functions is denoted as \( g \circ f \) (read as "g of f") which maps elements from set A to set C.

2. **Statement 1 Analysis**:
The first statement claims that if \( g \circ f = f \circ g \), then \( f^{-1} = g \) or \( g^{-1} = f \).
- This is generally not true. The equality \( g \circ f = f \circ g \) does not imply that the functions are inverses of each other.
- In fact, \( g \circ f = f \circ g \) means that the functions commute, which is a different property.

3. **Statement 2 Analysis**:
The second statement asserts that \( g \circ f \neq f \circ g \).
- To demonstrate this, we can use specific functions. Let \( f(x) = x + 1 \) and \( g(x) = x^2 \).
- Calculate \( g \circ f(x) \):
\[ g(f(x)) = g(x + 1) = (x + 1)^2 = x^2 + 2x + 1 \]
- Now calculate \( f \circ g(x) \):
\[ f(g(x)) = f(x^2) = x^2 + 1 \]
- Clearly, \( g(f(x)) \neq f(g(x)) \) since \( x^2 + 2x + 1 \neq x^2 + 1 \) for most values of \( x \).

4. **Conclusion**:
- From the analysis, we conclude that both statements are true:
- Statement 1 is true in that \( g \circ f = f \circ g \) does not imply \( f^{-1} = g \) or \( g^{-1} = f \).
- Statement 2 is also true as we have shown \( g \circ f \neq f \circ g \) with specific functions.

### Final Conclusion
Both statements are true, but they are not directly related to each other. Statement 2 does not provide a correct explanation for Statement 1.