Question

The domain and range for ${\text{sinh}}^{-1}x$ is

• A. R - {0}
• B. R- {1}
• C. R
• D. None of these

Answer 1

Answer: (C)

R

$\sinh x=\frac{e^x-e^{-x}}{2}$

$\because \sinh x$ is satisfying for all values of $x$

Hence $x\in R$

Now, $\sinh x=\frac{e^{2x}-1}{2e^x}$

Let, $e^x=t$

$\Rightarrow \frac{t^2-1}{2t}=\sinh x=y$

$\Rightarrow t^2-1=2ty$

$\Rightarrow t^2-2ty-1=0$

$\because t>0$

$4y^2+4>0$

$y^2+1>0$

which is true for all value.

Hence $y\in R$