The graphs and properties such as domain, range and asymptotes of the 6 hyperbolic functions: sinh(x), cosh(x),
tanh(x), coth(x), sech(x) and csch(x) are presented. The six hyperbolic functions are defined as follows:
Hyperbolic Sine Function : \( \sinh(x) = \dfrac{e^x - e^{-x}}{2} \)
The graph of \( y = \sinh(x) \) is shown below along with the graphs of \( y = \dfrac{e^x}{2} \) and \( y = - \dfrac{e^{-x}}{2} \) for comparison
Domain: \[ (-\infty , \infty) \]
Range: \[ (-\infty , \infty) \]
Odd function since: \(\sinh(-x) = - \sinh(x) \)
Fig.1 - Graph of Hyperbolic Sine Function sinh(x)
Hyperbolic Cosine Function : \( \cosh(x) = \dfrac{e^x + e^{-x}}{2} \)
The graph of \( y = \cosh(x) \) is shown below along with the graphs of \( y = \dfrac{e^x}{2} \) and \( y = \dfrac{e^{-x}}{2} \) for comparison
Domain: \[ (-\infty , \infty) \]
Range: \[ [1 , \infty) \]
Even function since: \(\sinh(-x) = \sinh(x) \)
Fig.2 - Graph of Hyperbolic Cosine Function cosh(x)
Hyperbolic Tangent Function : \( \tanh(x) = \dfrac{\sinh(x)}{\cosh(x)} = \dfrac{e^x - e^{-x}}{e^x + e^{-x}} \)
Domain: \[ (-\infty , \infty) \]
Range: \[ (-1 , 1) \]
Odd function since: \( \tanh(-x) = - \tanh(x) \} \)
Horizontal asymptote: \( y = 1 \) as \( x \) increases indefinitely
Horizontal asymptote: \( y = - 1 \) as \( x \) decreases indefinitely
Odd function since: \(\tanh(-x) = - \tanh(x) \)
Fig.3 - Graph of Hyperbolic Tangent Function tanh(x)
Hyperbolic Cotangent Function : \( \coth(x) = \dfrac{\cosh(x)}{\sinh(x)} = \dfrac{e^x + e^{-x}}{e^x - e^{-x}} \)
Domain: \[ (-\infty , 0) \cup (0, \infty) \]
Range: \[ (-\infty , -1) \cup (1 , \infty) \]
Odd function since: \( \coth(-x) = - \coth(x) \)
Horizontal asymptote: \( y = 1 \) as \( x \) increases indefinitely
Horizontal asymptote: \( y = - 1 \) as \( x \) decreases indefinitely
Vertical asymptote: \[ x = 0 \]
Odd function since: \(\coth(-x) = - \coth(x) \)
Fig.4 - Graph of Hyperbolic Cotangent Function coth(x)
Hyperbolic Secant Function : \( \text{sech}(x) = \dfrac{1}{\cosh(x)} = \dfrac{2}{e^x + e^{-x}} \)
Domain: \[ (-\infty , \infty) \]
Range: \[ (0 , 1] \]
Even function since: \( \text{sech}(-x) = \text{sech}(x) \)
Horizontal asymptote: \( y = 0 \) as \( x \) increases or decreases indefinitely
Fig.5 - Graph of Hyperbolic Secant Function sech(x)
Hyperbolic Cosecant Function : \( \text{csch}(x) = \dfrac{1}{\sinh(x)} = \dfrac{2}{e^x - e^{-x}} \)
Domain: \[ (-\infty , 0) \cup (0 , \infty) \]
Range: \[ (-\infty , 0) \cup (0 , \infty) \]
Odd function since : \[ \text{csch}(-x) = - \text{csch}(x) \]
Horizontal asymptote: \( y = 0 \) as \( x \) increases or decreases indefinitely
Vertical asymptote: \( x = 0 \)
Fig.6 - Graph of Hyperbolic Cosecant Function csch(x)
More references on hyperbolic functions