What is the inverse of #f(x)= -ln(arctan(x))# ?

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Answer 1 · 2015-11-13T05:53:40

#f^-1(x) = tan(e^-x)#

A typical way of finding an inverse function is to set #y = f(x)# and then solve for #x# to obtain #x = f^-1(y)#

Applying that here, we start with
#y = -ln(arctan(x))#

#=> -y = ln(arctan(x))#

#=>e^-y = e^(ln(arctan(x))) = arctan(x)# (by the definition of #ln#)

#=> tan(e^-y) = tan(arctan(x)) = x# (by the definition of #arctan#)

Thus we have #f^-1(x) = tan(e^-x)#

If we wish to confirm this via the definition #f^-1(f(x)) = f(f^-1(x)) = x#
remember that #y = f(x)# so we already have
#f^-1(y) = f^-1(f(x)) = x#

For the reverse direction,

#f(f^-1(x)) = -ln(arctan(tan(e^-x))#

#=> f(f^-1(x)) = -ln(e^-x)#

#=> f(f^-1(x)) = -(-x*ln(e)) = -(-x*1)#

#=> f(f^-1(x)) = x#