This is a mathematical problem set dealing with function analysis, differentiation, inverse functions, and integration. Here’s a breakdown of the key topics covered:

1. Function Analysis and Differentiation:

   • Studying the differentiability of $f(x) = \sqrt{\frac{x-1}{4x}}$ at $x = 1$.
   • Creating a variation table.
   • Plotting the graph of $f(x)$.

2. Inverse Function Analysis:

   • Proving that $f$ is bijective from $\left[ 1, \frac{4}{3} \right]$ onto $\left[ 0, \frac{1}{4} \right]$.
   • Deriving the inverse function $f^{-1}(x)$.
   • Plotting the inverse function’s graph.

3. Integral and Area Calculation:

   • Proving that $f^{-1}(x) = \frac{1}{1 - 4x^2}$ for $x \in \left[ 0, \frac{1}{4} \right]$.
   • Evaluating the integral $\int_0^{1/4} f^{-1}(x)dx$.
   • Using this to compute the bounded area.

Would you like detailed solutions for specific parts? Here are five related questions:

1. How do you determine the differentiability of $f(x)$ at a point?
2. How do you establish that a function is bijective on a given interval?
3. What steps are involved in finding the inverse of a function?
4. How do you evaluate an integral involving a rational function?
5. How do you compute an area under a curve using integration?

Tip: When proving bijectivity, always check both injectivity (one-to-one) and surjectivity (onto).