Question

Domain of the real-valued function \(f(x) = \log(x^2 - 1) + x \, \coth^{-1}x\) is

• A. \(\mathbb{R}\)
• B. \((-1,1)\)
• C. \(\mathbb{R} - [-1,1]\)
• D. \([0,1)\)

Hint:

Check domain restrictions for logarithmic and inverse hyperbolic functions separately, then take their intersection.

Answer 1

The Correct Option is C

For the function to be defined: 1. \(\log(x^2 - 1)\) is defined \(\Rightarrow x^2 - 1>0 \Rightarrow x<-1 \text{ or } x>1\).
2. \(\coth^{-1}x\) is defined for \(|x|>1\).
So the intersection of both conditions gives: \(x \in (-\infty, -1) \cup (1, \infty)\).
Therefore, domain = \(\mathbb{R} - [-1,1]\).