Question

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

Answer 1

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 2

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.