Evaluating hyperbolic functions Evaluate each expression without using a calculator or state that the value does not exist. Simplify answers as much as possible.
j. sinh⁻¹ (e² − 1)/2e

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Recall the definition of the inverse hyperbolic sine function: \(\sinh^{-1}(x) = y\) means \(\sinh(y) = x\).

Set \(y = \sinh^{-1}\left(\frac{e^{2} - 1}{2e}\right)\), so that \(\sinh(y) = \frac{e^{2} - 1}{2e}\).

Use the definition of hyperbolic sine: \(\sinh(y) = \frac{e^{y} - e^{-y}}{2}\), so write the equation as \(\frac{e^{y} - e^{-y}}{2} = \frac{e^{2} - 1}{2e}\).

Multiply both sides by 2 to clear denominators: \(e^{y} - e^{-y} = \frac{e^{2} - 1}{e}\).

Recognize that \(\frac{e^{2} - 1}{e} = e - e^{-1}\), so the equation becomes \(e^{y} - e^{-y} = e - e^{-1}\). From this, deduce that \(y = 1\) because \(e^{1} - e^{-1} = e - e^{-1}\).

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Inverse Hyperbolic Sine Function (sinh⁻¹)

The inverse hyperbolic sine function, sinh⁻¹(x), returns the value whose hyperbolic sine is x. It can be expressed as ln(x + √(x² + 1)), allowing evaluation without a calculator by simplifying the expression inside the logarithm.

Simplification of Algebraic Expressions

Simplifying algebraic expressions involves factoring, expanding, and reducing terms to their simplest form. This skill is essential to rewrite complex expressions like (e² − 1)/2e into a form that makes applying inverse hyperbolic functions easier.

Properties of Exponential Functions

Exponential functions with base e have properties such as e^a * e^b = e^(a+b) and (e^a)/(e^b) = e^(a-b). Understanding these helps in manipulating expressions involving e, which is crucial for simplifying and evaluating hyperbolic function arguments.

Evaluating hyperbolic functions Evaluate each expression without using a calculator or state that the value does not exist. Simplify answers as much as possible.j. sinh⁻¹ (e² − 1)/2e