Prove That The Function F(x)=tanh^-1(x) Is An Odd Function ? | Socratic

Prove that the function f(x)=tanh^-1(x) is an odd function ?

prove that the function f(x)=tanh^-1(x) is an odd function

Calculus

2 Answers

Ananda Dasgupta Apr 19, 2018

The argument below can be adapted to prove that the inverse of any odd invertible function is odd.

Explanation:

The function #tanh(x) equiv (e^x-e^-x)/(e^x+e^-x)# is obviously an odd function. So

#tanh(-y) = -tanh(y)#

Writing #tanh (y) = x#, or equivalently #y = tanh^-1(x)#, this equation becomes

#tanh(-tanh^-1(x))=-x#

This implies

#-tanh^-1(x) =tanh^-1(-x)#

(where we have used #tanh^-1(tanh(x))=x#)

So, #tanh^-1(-x) = -tanh^-1(x)# - and thus #tanh(x)# is an odd function.

Answer link Yahia M. Apr 19, 2018

See the explanation below

Explanation:

The logarithmic form of the function

#f(x)=# #tanh^-1x=1/2ln((1+x)/(1-x))#

Substitute each #x# by #-x#

#f(-x)=1/2ln((1-x)/(1+x))#

Using properties of logarithmic functions

#color(green) (ln(a/b)=lna-lnb)#

#=1/2(ln(1-x)-ln(1+x))#

take #-1# as a common factor

#=-1/2(ln(1+x)-ln(1-x))#

#=-1/2ln((1+x)/(1-x))=-f(x)#

#f(-x)=-f(x)#

#f(x)# is an odd function.

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