We are tasked with finding the range of the function:

$f(x) = \tan^{-1}(\sqrt{x} + [-x]) + \sqrt{2} - |x| + \frac{1}{x^2}$

Let's break it down step by step:

1. Inverse tangent (arctan): $\tan^{-1}(y)$ has a range of $(-\frac{\pi}{2}, \frac{\pi}{2})$ for all real values of $y$.

2. Analysis of $\sqrt{x} + [-x]$: The square root function, $\sqrt{x}$, is defined for $x \geq 0$. Therefore, this function is undefined for negative values of $x$, limiting the domain of $f(x)$ to $x \geq 0$.

3. Behavior of $\sqrt{x} + [-x]$ for non-negative $x$:

   • For $x \geq 0$, $[-x] = 0$ (since the greatest integer less than or equal to $0$ is 0). So the expression simplifies to just $\tan^{-1}(\sqrt{x})$.

   The range of $\tan^{-1}(\sqrt{x})$ as $x$ varies from $0$ to $\infty$ is $[0, \frac{\pi}{2})$.

4. Additional terms:

   • $\sqrt{2}$ is constant and equals approximately 1.414.
   • $|x|$ for $x \geq 0$ is just $x$, so the term $-|x|$ becomes $-x$.
   • The term $\frac{1}{x^2}$ is always positive for $x > 0$ and tends to 0 as $x \to \infty$, but becomes very large as $x \to 0^+$.

Considerations for the range:

• The term $\tan^{-1}(\sqrt{x})$ is bounded by $[0, \frac{\pi}{2})$.
• $\sqrt{2}$ shifts the overall function upward by 1.414.
• The term $-x$ decreases as $x$ increases.
• $\frac{1}{x^2}$ increases rapidly as $x$ approaches 0, adding a very large positive value near $x = 0$.

Behavior at the boundaries:

• As $x \to 0^+$:

  • $\tan^{-1}(\sqrt{x}) \to 0$,
  • $\sqrt{2} = 1.414$,
  • $-|x| \to 0$,
  • $\frac{1}{x^2} \to \infty$.

  Therefore, as $x$ approaches 0, $f(x) \to \infty$.

• As $x \to \infty$:

  • $\tan^{-1}(\sqrt{x}) \to \frac{\pi}{2}$,
  • $\sqrt{2} = 1.414$,
  • $-|x| \to -\infty$,
  • $\frac{1}{x^2} \to 0$.

  So, as $x$ grows large, $f(x) \to -\infty$.

Conclusion:

The function $f(x)$ tends to $\infty$ as $x \to 0^+$ and $-\infty$ as $x \to \infty$. Hence, the range of the function is:

$\boxed{(-\infty, \infty)}$

Do you need further details or explanations on any step?

Here are five related questions:

1. What is the range of $f(x) = \tan^{-1}(x) + \sin(x)$?
2. How do you find the domain of a function involving square roots?
3. What is the range of the function $g(x) = \frac{1}{x^2}$?
4. What properties does the function $\tan^{-1}(x)$ have?
5. What are the critical points of $f(x) = -|x| + \frac{1}{x^2}$?

Tip: When working with inverse trigonometric functions, it’s useful to remember their principal value ranges!