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The hyperbolic tangent function is a mathematical function that maps real numbers to values between -1 and 1.

The hyperbolic tangent function, denoted by tanh(x), is defined as:

tanh(x) = (e^x - e^-x) / (e^x + e^-x)

where e is the mathematical constant approximately equal to 2.71828.

The graph of the hyperbolic tangent function is similar to that of the standard tangent function, but it is symmetric about the y-axis and asymptotic to y = -1 and y = 1 as x approaches negative and positive infinity, respectively.

The hyperbolic tangent function is used in various areas of mathematics and science, including calculus, differential equations, and physics. It is particularly useful in modelling systems that exhibit saturation or limiting behaviour, such as the response of a neuron in the brain to a stimulus.

Some important properties of the hyperbolic tangent function include:

- tanh(x) is an odd function, meaning that tanh(-x) = -tanh(x)

- tanh(x) is continuous and differentiable for all real numbers x

- tanh(x) is strictly increasing on the interval (-infinity, 0) and strictly decreasing on the interval (0, infinity)

- tanh(x) has a horizontal asymptote of y = -1 as x approaches negative infinity and a horizontal asymptote of y = 1 as x approaches positive infinity.

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