The hyperbolic tangent function is defined by tanh(x)= (e^x-e^-x) /(e^x+e^-x) (a) Prove that tan

step 1 First thing we need to do is evaluate sine hX over cosine HX. And this equals sine hX times 1 ov

The hyperbolic tangent function is defined by tanh(x)=...

The hyperbolic tangent function is defined by $\tanh (x)=$ $\left(e^{x}-e^{-x}\right) /\left(e^{x}+e^{-x}\right)$
(a) Prove that $\tanh (x)=\sinh (x) / \cosh (x)$
(b) Prove that tanh is an odd function.
(c) Prove that $f(x)=1+\tanh (x)$ is a logistic function.

Key Concepts

Logistic Function

A logistic function typically has the form L/(1 + e^{-k(x-x0)}) and models growth that starts exponentially and then saturates to a maximum value. Demonstrating that f(x) = 1 + tanh(x) can be rewritten into this form shows how hyperbolic functions are closely related to logistic functions, linking the algebraic manipulation of hyperbolic identities to applications in statistics, biology, and other fields where growth processes are modeled.

Odd and Even Functions

Understanding odd and even functions is crucial in analyzing function symmetry. A function is odd if f(-x) = -f(x) and even if f(-x) = f(x). With hyperbolic functions, recognizing that sinh is odd and cosh is even immediately implies that tanh, as the ratio of an odd and an even function, is also odd, which is used to prove function symmetry properties.

Hyperbolic Functions

Hyperbolic functions, such as sinh, cosh, and tanh, are defined using exponential functions. They parallel trigonometric functions but for a hyperbola rather than a circle. Their definitions using exponentials provide useful identities and properties that allow for algebraic manipulation, such as proving relationships between them or transforming them into other functional forms.

Relationship Between Hyperbolic Functions

The relationship tanh(x) = sinh(x)/cosh(x) is a key identity among hyperbolic functions. It is derived directly from the definitions of sinh and cosh in terms of exponential functions, emphasizing how composite functions can be expressed in simpler or alternative forms that illuminate their properties and simplify proofs.

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