Statement-1: If `f:R to R and g:R to R ` be two functions such that `f(x)=x^(2) and g(x)=x^(3)`, then fog (x)=gof (x).
Statement-2: The composition of functions is commulative.

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem, we need to evaluate the two statements regarding the functions \( f \) and \( g \).

### Given:
- \( f(x) = x^2 \)
- \( g(x) = x^3 \)

### Statement 1:
We need to check if \( f(g(x)) = g(f(x)) \).

1. **Calculate \( f(g(x)) \)**:
\[ f(g(x)) = f(x^3) = (x^3)^2 = x^6 \]

2. **Calculate \( g(f(x)) \)**:
\[ g(f(x)) = g(x^2) = (x^2)^3 = x^6 \]

3. **Compare the results**:
Since both \( f(g(x)) \) and \( g(f(x)) \) yield \( x^6 \), we have:
\[ f(g(x)) = g(f(x)) \]

Thus, Statement 1 is **True**.

### Statement 2:
We need to check if the composition of functions is commutative, i.e., if \( f(g(x)) = g(f(x)) \) for all functions \( f \) and \( g \).

To check this, we can consider different functions. Let's take:
- \( f(x) = x^2 \)
- \( g(x) = \sin(x) \)

1. **Calculate \( f(g(x)) \)**:
\[ f(g(x)) = f(\sin(x)) = (\sin(x))^2 \]

2. **Calculate \( g(f(x)) \)**:
\[ g(f(x)) = g(x^2) = \sin(x^2) \]

3. **Compare the results**:
\[ f(g(x)) = \sin^2(x) \quad \text{and} \quad g(f(x)) = \sin(x^2) \]
Since \( \sin^2(x) \neq \sin(x^2) \), we conclude that:
\[ f(g(x)) \neq g(f(x)) \]

Thus, Statement 2 is **False**.

### Final Conclusion:
- Statement 1 is **True**.
- Statement 2 is **False**.

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