Find the area bounded by `y=f(x)` and the curve `y=2/(1+x^2)` satisfying the condition `f(x),f(y)=f(xy) AA x,y in R and f'(1)=2,f(1)=1`,

Answer

To find the area bounded by the curve \( y = f(x) \) and the curve \( y = \frac{2}{1+x^2} \), we need to determine the function \( f(x) \) based on the given conditions and then calculate the area between the two curves.

### Step 1: Determine the function \( f(x) \)

We are given that \( f(x) \) satisfies the condition \( f(x) f(y) = f(xy) \) for all \( x, y \in \mathbb{R} \). A common function that satisfies this condition is of the form \( f(x) = x^a \).