Functional Equation

f is a function such as

• $f:[0,\mathrm{\infty }[\to \mathbf{R}$
• $\text{f is strictly increasing}$
• $f(x)>-\frac{1}{x}$
• $f(x)\times f(f(x)+\frac{1}{x})=1\text{ for all x}>0$

What is f(1)?

Give your answer rounded to 10 digits after the decimal point.