Inverse Hyperbolic Tangent is Odd Function/Proof 1

From ProofWiki
Jump to navigation Jump to search

Theorem

$\map {\tanh^{-1} } {-x} = -\tanh^{-1} x$


Proof

\(\ds \map {\tanh^{-1} } {-x}\) \(=\) \(\ds y\)
\(\ds \leadstoandfrom \ \ \) \(\ds -x\) \(=\) \(\ds \tanh y\) Definition 1 of Inverse Hyperbolic Tangent
\(\ds \leadstoandfrom \ \ \) \(\ds x\) \(=\) \(\ds -\tanh y\)
\(\ds \leadstoandfrom \ \ \) \(\ds x\) \(=\) \(\ds \map \tanh {-y}\) Hyperbolic Tangent Function is Odd
\(\ds \leadstoandfrom \ \ \) \(\ds \tanh^{-1} x\) \(=\) \(\ds -y\) Definition 1 of Inverse Hyperbolic Tangent

$\blacksquare$