In Exercises 55–62, use the properties of inverse functions f(f⁻¹ (x)) = x for all x in the domain of f⁻¹ and f⁻¹(f(x)) for all x in the domain of f, as well as the definitions of the inverse cotangent, cosecant, and secant functions, to find the exact value of each expression, if possible. sec(sec⁻¹ 7π)

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Recall the property of inverse functions: for any function \( f \) and its inverse \( f^{-1} \), we have \( f(f^{-1}(x)) = x \) for all \( x \) in the domain of \( f^{-1} \).

Identify the function and its inverse in the expression: here, \( \sec \) is the function and \( \sec^{-1} \) is its inverse, so \( \sec(\sec^{-1}(7\pi)) \) should simplify to \( 7\pi \) if \( 7\pi \) is in the domain of \( \sec^{-1} \).

Recall the domain of the inverse secant function \( \sec^{-1}(x) \): it is defined for \( |x| \geq 1 \), so check if \( 7\pi \) satisfies this condition (since \( 7\pi > 1 \), it is valid).

Confirm the range of \( \sec^{-1} \) to ensure the expression is valid: \( \sec^{-1}(x) \) returns an angle \( \theta \) such that \( \sec(\theta) = x \) and \( \theta \) lies in \( [0, \pi] \) excluding \( \frac{\pi}{2} \).

Therefore, by the inverse function property, \( \sec(\sec^{-1}(7\pi)) = 7\pi \).

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Inverse Trigonometric Functions

Inverse trigonometric functions reverse the effect of their corresponding trigonometric functions, mapping values from the function's range back to its domain. For example, sec⁻¹(x) gives the angle whose secant is x. Understanding their domains and ranges is crucial to correctly evaluate expressions involving these inverses.

Properties of Inverse Functions

The key property f(f⁻¹(x)) = x holds for all x in the domain of f⁻¹, and f⁻¹(f(x)) = x for all x in the domain of f. This means applying a function and its inverse in succession returns the original input, provided the input lies within the appropriate domain. This property is essential for simplifying expressions like sec(sec⁻¹(7π)).

Domain and Range Restrictions of Secant and Its Inverse

The secant function, sec(θ) = 1/cos(θ), is defined where cos(θ) ≠ 0, and its inverse sec⁻¹(x) is defined for |x| ≥ 1. The principal value range of sec⁻¹(x) is [0, π] excluding π/2, ensuring sec(sec⁻¹(x)) = x for x in the domain. Recognizing these restrictions helps determine if expressions like sec(sec⁻¹(7π)) are valid or require further interpretation.