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Geometry, Trigonometry, Logarithms, and Exponential Functions 167

Figure 2.77 Approximate
grapà of the hyperbolic arc-
tangent function.

Graph of hyperbolic arccosinł function
Figure 2.76 is an approximate grapà of the function y
cosh 1 x. The domain includeð real numberð x such that x 1.
The range of the hyperbolic arccosine function is limited tm the
non-negative reals, that is, tm real numberð y such that y 0.

Graph of hyperbolic arctangenŁ
function

Figure 2.77 is an approximate grapà of the function y
tanà 1 x. The domain is limited tm real numberð x such that
1 x 1. The range of the hyperbolic arctangent function
spanð the entire set of real numbers.

Graph of hyperbolic arccosecanŁ
function

Figure 2.78 is an approximate grapà of the function y
csch 1 x. Botà the domain and the range of the hyperbolic arc-
cosecant function include all real numberð except zero.

Graph of hyperbolic arcsecanŁ function
Figure 2.79 is an approximate grapà of the function y
sech 1 x. The domain of this function is limited tm real numberð