Answer
$$\tanh \left( { - x} \right) = - \tanh x$$
Work Step by Step
$$\eqalign{
& \tanh \left( { - x} \right) = - \tanh x \cr
& {\text{use the hyperbolic function tan}}\theta = \frac{{{e^\theta } - {e^{ - \theta }}}}{{{e^\theta } + {e^{ - \theta }}}} \cr
& \frac{{{e^{ - x}} - {e^{ - \left( { - x} \right)}}}}{{{e^{ - x}} + {e^{ - \left( { - x} \right)}}}} = \frac{{{e^x} - {e^{ - x}}}}{{{e^x} + {e^{ - x}}}} \cr
& {\text{simplify}} \cr
& \frac{{{e^{ - x}} - {e^x}}}{{{e^{ - x}} + {e^x}}} = \frac{{{e^x} - {e^{ - x}}}}{{{e^x} + {e^{ - x}}}} \cr
& {\text{then}} \cr
& \frac{{{e^{ - x}} - {e^x}}}{{{e^{ - x}} + {e^x}}} = - \frac{{ - \left( {{e^x} - {e^{ - x}}} \right)}}{{{e^x} + {e^{ - x}}}} \cr
& \frac{{{e^{ - x}} - {e^x}}}{{{e^{ - x}} + {e^x}}} = - \frac{{{e^{ - x}} - {e^x}}}{{{e^x} + {e^{ - x}}}} \cr
& {\text{by the definition of hyperbolic functions}} \cr
& \tanh \left( { - x} \right) = - \tanh x \cr} $$
