Consider the function f: R → R defined by f(x)=(2 x)/(√(1+9 x^2)). If the composition of f. (f ° f ° f ° ⋯ ° f)_10 times (x)=(2^10 x)/(√(1+9 α x^2)),…

Consider the function $\displaystyle f: \mathbf{R} \rightarrow \mathbb{R}$ defined by $\displaystyle f(x)=\frac{2 x}{\sqrt{1+9 x^2}}$. If the composition of f. $\displaystyle \underbrace{(f \circ f \circ f \circ \cdots \circ f)}_{10 \text { times }}(x)=\frac{2^{10} x}{\sqrt{1+9 \alpha x^2}}$, then the value of $\displaystyle \sqrt{3 \alpha+1}$ is equal to $\displaystyle \_\_\_\_$