Question
Write the range of:
\[ \tan^{-1}x \]
Solution
The inverse tangent function \( \tan^{-1}x \) is defined for all real values of \( x \).
Its principal value range is:
\[ \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \]
This means:
- \( \tan^{-1}x \) never equals \( \frac{\pi}{2} \) or \( -\frac{\pi}{2} \)
- But it approaches these values as \( x \to \infty \) or \( x \to -\infty \)
Final Answer:
\[ \boxed{ (-\frac{\pi}{2}, \frac{\pi}{2}) } \]
Key Concept
Inverse tangent is restricted to this interval to make it a one-to-one function.