 Range of the function f(x)=tan-1(√[x]+[-x])+√2-|x| +1/x² is

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!

Range of the function f(x) = tan^(-1)(√[x] + [-x]) + √2 - |x| + 1/x² is

Explore how to determine the range of the function f(x) = tan^(-1)(√x + [-x]) + √2 - |x| + 1/x². The solution includes key concepts such as inverse trigonometric functions, square roots, and asymptotic behavior. Suitable for advanced high school or early college students.

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