Standard +0.8 This is a standard Further Maths proof of an inverse hyperbolic function's logarithmic form, requiring manipulation of exponentials and logarithms, followed by a non-routine application requiring substitution and algebraic manipulation. The proof itself is bookwork, but part (b) requires insight to apply the result and handle the domain restrictions carefully.
2. (a) Prove that
$$\tanh ^ { - 1 } ( x ) = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right) \quad - k < x < k$$
stating the value of the constant \(k\).
(b) Hence, or otherwise, solve the equation
$$2 x = \tanh ( \ln \sqrt { 2 - 3 x } )$$