Question

Let the function \( g: (-\infty, 0) \rightarrow \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \) be given by \( g(u) = 2 \tan^{-1}(e^u) - \frac{\pi}{2} \). Determine the properties of \( g \).

• A. Even and is strictly increasing in \( (0, \infty) \)
• B. Odd and is strictly decreasing in \( (-\infty, 0) \)
• C. Odd and is strictly increasing in \( (-\infty, \infty) \)
• D. Neither even nor odd, but is strictly increasing in \( (-\infty, \infty) \)

Hint:

Remember, for a function to be odd, \( f(-x) = -f(x) \) must hold true, and the function's derivative should be positive for increasing nature.

Answer 1

The Correct Option is C

Step 1: {Analyze evenness or oddness}
\[ g(-u) = 2 \tan^{-1}(e^{-u}) - \frac{\pi}{2} \] \[ = 2 \left( \frac{\pi}{2} - \tan^{-1}(e^u) \right) - \frac{\pi}{2} \] \[ = \pi - 2 \tan^{-1}(e^u) - \frac{\pi}{2} = -2 \tan^{-1}(e^u) + \frac{\pi}{2} \] \[ = -(2 \tan^{-1}(e^u) - \frac{\pi}{2}) = -g(u) \] Thus, \( g \) is an odd function. 
Step 2: {Verify increasing nature}
The derivative \( g'(u) = 2\frac{1}{1+e^{2u}}e^u>0 \) for all \( u \), indicating \( g \) is strictly increasing over \( (-\infty, \infty) \).