What is the domain of sech⁻¹ x? How is sech⁻¹ x defined in terms of the inverse hyperbolic cosine?

Verified step by step guidance

1. Step 1: Recall the definition of the hyperbolic secant function, sech(x), which is given by sech(x) = 1 / cosh(x), where cosh(x) is the hyperbolic cosine function.
2. Step 2: Understand that the inverse hyperbolic secant function, sech⁻¹(x), is defined for values of x within the domain of the hyperbolic secant function. Since sech(x) outputs values in the range (0, 1], the domain of sech⁻¹(x) is x ∈ (0, 1].
3. Step 3: Recognize that sech⁻¹(x) can be expressed in terms of the inverse hyperbolic cosine function, cosh⁻¹(x). Specifically, sech⁻¹(x) = cosh⁻¹(1/x), where 1/x must be greater than or equal to 1 to ensure the argument of cosh⁻¹ is valid.
4. Step 4: Verify that the domain of sech⁻¹(x) aligns with the range of sech(x). Since sech(x) is defined for all real numbers and outputs values in (0, 1], the domain of sech⁻¹(x) is restricted to x ∈ (0, 1].
5. Step 5: Conclude that the domain of sech⁻¹(x) is x ∈ (0, 1], and it is defined in terms of the inverse hyperbolic cosine as sech⁻¹(x) = cosh⁻¹(1/x).

Key Concepts

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the inverse hyperbolic secant function, sech⁻¹ x, it is essential to determine the values of x that yield valid outputs. Understanding the domain helps in identifying the range of the function and ensures that calculations are performed within valid limits.

Inverse Hyperbolic Functions

Inverse hyperbolic functions are the inverses of hyperbolic functions, similar to how inverse trigonometric functions relate to trigonometric functions. The function sech⁻¹ x is defined as the inverse of the hyperbolic secant function, which is related to the hyperbolic cosine. This relationship allows us to express sech⁻¹ x in terms of the inverse hyperbolic cosine, specifically as sech⁻¹ x = cosh⁻¹(1/x) for x in the appropriate domain.

Hyperbolic Functions

Hyperbolic functions, such as sinh, cosh, and sech, are analogs of trigonometric functions but are based on hyperbolas rather than circles. The hyperbolic secant function, sech(x), is defined as 1/cosh(x), where cosh(x) is the hyperbolic cosine. Understanding these functions is crucial for grasping the properties and definitions of their inverses, including sech⁻¹ x, and how they relate to real numbers and their geometric interpretations.