Graphs of Hyperbolic Functions: Domain, Range, and Asymptotes

The graphs and properties—such as domain, range, and asymptotes—of the 6 hyperbolic functions (\(\sinh(x)\), \(\cosh(x)\), \(\tanh(x)\), \(\coth(x)\), \(\text{sech}(x)\), and \(\text{csch}(x)\)) are presented below[cite: 1]. These functions are defined as follows[cite: 1]:

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: \((-\infty, \infty)\)[cite: 1]
Range: \((-\infty, \infty)\)[cite: 1]
Symmetry: Odd function since \(\sinh(-x) = - \sinh(x)\)[cite: 1]

Graph of hyperbolic sine function 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: \((-\infty, \infty)\)[cite: 1]
Range: \([1, \infty)\)[cite: 1]
Symmetry: Even function since \(\cosh(-x) = \cosh(x)\)

Graph of hyperbolic cosine function cosh(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: \((-\infty, \infty)\)[cite: 1]
Range: \((-1, 1)\)[cite: 1]
Symmetry: Odd function since \(\tanh(-x) = - \tanh(x)\)[cite: 1]
Horizontal Asymptotes: \( y = 1 \) as \( x \) increases indefinitely[cite: 1]; \( y = -1 \) as \( x \) decreases indefinitely[cite: 1].

Graph of hyperbolic tangent function 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: \((-\infty, 0) \cup (0, \infty)\)[cite: 1]
Range: \((-\infty, -1) \cup (1, \infty)\)[cite: 1]
Symmetry: Odd function since \(\coth(-x) = - \coth(x)\)[cite: 1]
Asymptotes: Horizontal asymptotes at \( y = 1 \) (as \( x \to \infty \)) and \( y = -1 \) (as \( x \to -\infty \))[cite: 1]; Vertical asymptote at \( x = 0 \)[cite: 1].

Graph of hyperbolic cotangent function 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: \((-\infty, \infty)\)[cite: 1]
Range: \((0, 1]\)[cite: 1]
Symmetry: Even function since \(\text{sech}(-x) = \text{sech}(x)\)[cite: 1]
Horizontal Asymptote: \( y = 0 \) as \( x \) increases or decreases indefinitely[cite: 1].

Graph of hyperbolic secant function sech(x) — Fig. 5 - Graph of Hyperbolic Secant Function \(\text{sech}(x)\)

Hyperbolic Cosecant Function: \[ \text{csch}(x) = \dfrac{1}{\sinh(x)} = \dfrac{2}{e^x - e^{-x}} \]

Domain: \((-\infty, 0) \cup (0, \infty)\)[cite: 1]
Range: \((-\infty, 0) \cup (0, \infty)\)[cite: 1]
Symmetry: Odd function since \(\text{csch}(-x) = - \text{csch}(x)\)[cite: 1]
Asymptotes: Horizontal asymptote at \( y = 0 \) as \( x \) increases/decreases indefinitely[cite: 1]; Vertical asymptote at \( x = 0 \)[cite: 1].

Graph of hyperbolic cosecant function csch(x) — Fig. 6 - Graph of Hyperbolic Cosecant Function \(\text{csch}(x)\)

Graphs of Hyperbolic Functions