The expression shown in the image is:

$f.H(x) = \frac{\sqrt{x^2 - 4}}{x^2 + 1}$

Steps to Analyze the Function:

1. Domain:

   • The expression inside the square root, $x^2 - 4$, must be non-negative. Therefore, $x^2 - 4 \geq 0$ implies $x^2 \geq 4$. This gives $x \leq -2$ or $x \geq 2$.
   • The denominator $x^2 + 1$ is always positive and non-zero for all real values of $x$, so it doesn't impose any further restrictions.
   • Hence, the domain of $f.H(x)$ is $x \in (-\infty, -2] \cup [2, \infty)$.

2. Behavior at Critical Points:

   • Analyze the function at critical points $x = \pm 2$ and check the limits as $x$ approaches these points.

3. Asymptotic Behavior:

   • Consider the limits as $x \rightarrow \infty$ and $x \rightarrow -\infty$.

4. Simplifications:

   • Investigate if there are any algebraic simplifications or possible factorizations.

Would you like to proceed with a detailed analysis of these steps? Additionally, here are some questions to explore further:

1. What is the range of the function $f.H(x)$?
2. How does $f.H(x)$ behave as $x$ approaches $\pm \infty$?
3. What are the critical points of $f.H(x)$?
4. How does $f.H(x)$ change when $x = 0$?
5. What is the behavior of $f.H(x)$ near the boundaries of its domain?

Tip: Always start by determining the domain of a function before attempting to solve or simplify it. This helps in understanding the behavior of the function and avoiding errors in interpretation.