https://file.aiquickdraw.com/mathbot/file/17299260953329b2neek9.jpg

The question asks whether the following inequality is true or false:

\[
-\frac{\pi}{2} < \tan^{-1}(-x) < \frac{\pi}{2}
\]

### Solution

1. **Understanding the Range of \( \tan^{-1} \) (Arctan)**:
   - The function \( \tan^{-1}(y) \) (or \( \arctan(y) \)) has a range of \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).
   - This range means that for any real number \( y \), the output of \( \tan^{-1}(y) \) will always lie within \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \).

2. **Applying to \( \tan^{-1}(-x) \)**:
   - Since \( x \) is a real number, \( -x \) is also real, and so \( \tan^{-1}(-x) \) will also produce a value within the interval \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).

3. **Conclusion**:
   - The statement \( -\frac{\pi}{2} < \tan^{-1}(-x) < \frac{\pi}{2} \) is **true** because it directly follows from the range of the \( \tan^{-1} \) function.

Therefore, the answer is **True**.

Would you like further clarification or a more detailed explanation?

---

### Related Questions:
1. What is the domain and range of the \( \tan^{-1} \) function?
2. How does the \( \tan^{-1} \) function behave for positive vs. negative inputs?
3. What other inverse trigonometric functions have restricted ranges?
4. How would the inequality change if it involved \( \tan^{-1}(x) \) instead of \( \tan^{-1}(-x) \)?
5. What is the derivative of the \( \tan^{-1}(x) \) function, and how is it used?

### Tip:
Remember that inverse trigonometric functions have restricted ranges to ensure they are functions (one-to-one mappings). This is key when evaluating inequalities involving them.

Is the inequality \(-\frac{\pi}{2} < \tan^{-1}(-x) < \frac{\pi}{2}\) true or false?

Learn whether the inequality -π/2 < tan^(-1)(-x) < π/2 holds true, focusing on the range of the arctangent function. This problem explores inverse trigonometric functions and inequalities, suitable for high school math students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation