Question

What is the Domain and Range of inverse hyperbolic tangent function?

• A. Domain $(0,1)$ and Range $[0,\infty )$
• B. Domain $(-1,1)$ and Range $(-\infty ,\infty )$
• C. Domain $(-\infty ,-1)$ and Range $(-\infty ,\infty )$
• D. Domain $(1,\infty )$ and Range $[1,\infty )$

Answer 1

The correct option is B Domain $(-1,1)$ and Range $(-\infty ,\infty )$

${\tanh }^{-1}x=\frac{1}{2}\ln \left|\frac{1+x}{1-x}\right|$

$\because 1-x>0$

$x<1$

also, $\frac{1+x}{1-x}>0$

$\frac{x+1}{x-1}<0$

$\Rightarrow x\in (-1,1)$

For range,

$y=\frac{1}{2}\ln \frac{1+x}{1-x}$

$\Rightarrow 2y=\ln \frac{1+x}{1-x}$

$\Rightarrow e^{2y}=\frac{1+x}{1-x}$

$\Rightarrow (1-x)e^{2y}=1+x$

$\Rightarrow e^{2y}-x{e}^{2y}=1+x$

$e^{2y}-1=x(1+e^{2y})$

$\Rightarrow \frac{e^{2y}-1}{e^{2y}+1}=x$

It is satisfied for all values of $y$. Hence $y\in (-\infty ,\infty )$