Graph of the Hyperbolic Tangent $y = \tanh x$

The hyperbolic tangent function $y = \tanh x$ is defined as the ratio of the hyperbolic sine to the hyperbolic cosine:

It is the hyperbolic analogue of the ordinary tangent $\tan x$.

Domain and range

Because the denominator $\cosh x \geq 1$ is never zero, $\tanh x$ is defined for every real $x$, giving the domain $(-\infty, \infty)$. Its values always lie strictly between $-1$ and $1$, so its range is the open interval $(-1, 1)$.

Symmetry

Since $\tanh(-x) = -\tanh x$, the function is odd, and its graph is symmetric about the origin.

Monotonicity

The derivative is $\dfrac{d}{dx}\tanh x = \operatorname{sech}^2 x = 1 - \tanh^2 x$, which is always positive, so $\tanh x$ is strictly increasing on the whole real line. The slope is steepest ($=1$) at the origin and flattens toward both ends, producing a smooth S-shaped curve.

Asymptotes and limits

As $x \to +\infty$, $\tanh x \to 1$, and as $x \to -\infty$, $\tanh x \to -1$. The lines $y = 1$ and $y = -1$ are therefore horizontal asymptotes that the curve never crosses.

Notable points

The curve passes through the origin with $\tanh 0 = 0$. This is an inflection point and the center of symmetry. Near the origin $\tanh x \approx x$.

Relationships with other functions

By definition it is the ratio of $\sinh x$ and $\cosh x$. It can also be written $\tanh x = \dfrac{e^{2x} - 1}{e^{2x} + 1}$, and it is related to the logistic sigmoid $\sigma(x) = \dfrac{1}{1 + e^{-x}}$ by $\tanh x = 2\sigma(2x) - 1$.

Applications

Because it is smooth, bounded, and S-shaped, $\tanh x$ has been widely used as an activation function in neural networks. In statistics and physics it appears as a saturating function that confines values to a range, and as a model for magnetization or signal response.