Functions

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