Question

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

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

Answer 1

The correct option is D Domain $[1,\infty )$ and Range $[0,\infty )$

$\cosh x=\frac{e^x-e^{-x}}{2}$ at $x=0$, $\cosh x=1$ So, $x\in [0,\infty ]$ and range$=y\in [1,\infty )$

$\because$ A.M$\ge$G.M

$\Rightarrow \frac{e^x-e^{-x}}{2}\ge \sqrt{e^x-e^{-x}}$

$\Rightarrow e^x-e^{-x}\ge 2$

which is clearly seen by the graph of $\cosh x$