Hyperbolic Functions: Inverses - Metric

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Hyperbolic Functions: Inverses

The hyperbolic sine function, \sinh x, is one-to-one, and therefore has a well-defined inverse, \sinh^{-1} x, shown in blue in the figure. In order to invert the hyperbolic cosine function, however, we need (as with square root) to restrict its domain. By convention, \cosh^{-1} x is taken to mean the positive number y such that x=\cosh y. This function is shown in red in the figure; notice that \cosh^{-1} x is defined only for x\ge 1 (at least where real numbers are concerned).

Figure 1: Plots of span class='math'\cosh^{-1} x/span (red), span class='math'\sinh^{-1} x/span (blue) and /span\tanh^{-1}

Figure 1: Plots of \cosh^{-1} x (red), \sinh^{-1} x (blue) and \tanh^{-1} x (green)

The hyperbolic tangent function is also one-to-one and invertible; its inverse, \tanh^{-1} x, is shown in green. It is defined only for -1< x

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