Statement-1 : A function `f : R rarr R` satisfies the f(x) - f(y) = x-y, `AA x, y in R, f(0), f(1) gt 0`, then f(x) = x+1.
and
Statement-2 : If function `f : R rarr R` satisfied the above relationship then `f(1) + f(0) + f(3)` is equal to 7.

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 a correct explanation for Statement-1.
C. Statement -1 is False, Statement -2 is False
D. Statement -1 is False, Statement -2 is True

Answer

To solve the problem, we will analyze both statements step by step.

### Step 1: Analyze Statement 1
We are given that a function \( f: \mathbb{R} \to \mathbb{R} \) satisfies the equation:
\[ f(x) - f(y) = x - y \quad \forall x, y \in \mathbb{R} \]
This implies that the function \( f \) is linear.

### Step 2: Rearranging the Equation
From the equation \( f(x) - f(y) = x - y \), we can rearrange it to express \( f(x) \):
\[ f(x) = f(y) + (x - y) \]
This indicates that the difference in the function values is equal to the difference in the input values.

### Step 3: Finding the Form of the Function
Let’s assume \( f(x) = x + c \) for some constant \( c \). Then:
\[ f(x) - f(y) = (x + c) - (y + c) = x - y \]
This holds true for any constant \( c \).

### Step 4: Using Given Conditions
We know from the problem statement that \( f(0) > 0 \) and \( f(1) > 0 \):
- \( f(0) = 0 + c = c > 0 \)
- \( f(1) = 1 + c > 0 \)

From \( f(0) > 0 \), we conclude \( c > 0 \).

### Step 5: Specific Form of the Function
If we take \( c = 1 \), then:
\[ f(x) = x + 1 \]
This satisfies both conditions \( f(0) = 1 > 0 \) and \( f(1) = 2 > 0 \).

### Conclusion for Statement 1
Thus, we can conclude that:
\[ f(x) = x + 1 \]
is indeed a solution that satisfies the conditions given in Statement 1.

### Step 6: Analyze Statement 2
Now we need to check Statement 2:
\[ f(1) + f(0) + f(3) \]
Calculating each term:
- \( f(1) = 1 + 1 = 2 \)
- \( f(0) = 0 + 1 = 1 \)
- \( f(3) = 3 + 1 = 4 \)

Adding these:
\[ f(1) + f(0) + f(3) = 2 + 1 + 4 = 7 \]

### Conclusion for Statement 2
Thus, Statement 2 is also true.

### Final Conclusion
Both statements are true, but Statement 2 is not a correct explanation for Statement 1 since it simply verifies a calculation based on the function derived from Statement 1.

### Summary
- **Statement 1**: True (the function is \( f(x) = x + 1 \))
- **Statement 2**: True (but not a correct explanation for Statement 1)