# What is the domain and range of cos^-1(tanh x)?

The domain is $x \in \mathbb{R}$ the range is $y \in \mathbb{R} : 0 < y < \pi$
The domain of the $\tanh \left(x\right)$ function is any real number: $x \in \mathbb{R}$
The range of the $\tanh \left(x\right)$ function is $y \in \mathbb{R} : - 1 < y < 1$; this does not exceed the domain of the ${\cos}^{-} 1 \left(x\right)$ function but it will never be able to force it to the values $0 \mathmr{and} \pi$.
Therefore, the range of is : $y \in \mathbb{R} : 0 < y < \pi$