Question

If \(A\) and \(B\) are the entire domain and range, respectively, of the real-valued function \[ f(x)=\cos^{-1}\!\left(\frac{2-x^2}{2+x^2}\right), \] then \(A\cap B=\)

• A. \( [0, \sqrt{2}) \)
• B. \( [0, \frac{\pi}{2}) \)
• C. \( [0, \pi) \)
• D. \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \)

Hint:

Use the substitution \( x = \sqrt{2}\tan\theta \). Then \( \frac{2-x^2}{2+x^2} = \frac{2-2\tan^2\theta}{2+2\tan^2\theta} = \cos 2\theta \). This simplifies the function to \( f(x) = \cos^{-1}(\cos 2\theta) \), making it easier to visualize the range.

Answer 1

The Correct Option is C