Question Number 133903 by shaker last updated on 25/Feb/21
Commented by mr W last updated on 25/Feb/21
$${what}\:{do}\:{mean}\:{with}\:{argth}\left({x}\right)?\:{is}\:{it} \\ $$$${the}\:{same}\:{as}\:{arctanh}\left({x}\right)\:{in}\:{your} \\ $$$${country}? \\ $$
Commented by mr W last updated on 25/Feb/21
$${say}\:{u}=\mathrm{tanh}\:\left({x}\right) \\ $$$$\Rightarrow{x}={arc}\mathrm{tanh}\:\left({u}\right)\:\:\:{this}\:{is}\:{definition} \\ $$$${i}.{e}.\: \\ $$$$\Rightarrow{x}={arc}\mathrm{tanh}\:\left(\mathrm{tanh}\:\left({x}\right)\right) \\ $$$$ \\ $$$${for}\:{the}\:{same}\:{reason} \\ $$$$\mathrm{tanh}\:\left({arc}\mathrm{tanh}\:\left({x}\right)\right)={x} \\ $$
Commented by mr W last updated on 25/Feb/21
$${generally}\:\left({and}\:{usually}\right) \\ $$$${f}^{−\mathrm{1}} \left({f}\left({x}\right)\right)={x} \\ $$$${f}\left({f}^{−\mathrm{1}} \left({x}\right)\right)={x} \\ $$