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)

Verified step by step guidance

1. Identify the expression: \( \sec(\sec^{-1}(7\pi)) \).
2. Recall the property of inverse functions: \( f(f^{-1}(x)) = x \) for all \( x \) in the domain of \( f^{-1} \).
3. Recognize that \( \sec^{-1}(x) \) is the inverse of the secant function, meaning \( \sec(\sec^{-1}(x)) = x \) for all \( x \) in the domain of \( \sec^{-1} \).
4. Determine the domain of \( \sec^{-1}(x) \), which is \( |x| \geq 1 \).
5. Since \( 7\pi \) is greater than 1, it is within the domain of \( \sec^{-1} \), so \( \sec(\sec^{-1}(7\pi)) = 7\pi \).

Key Concepts

Inverse Functions

Inverse functions are functions that reverse the effect of the original function. For a function f and its inverse f⁻¹, the property f(f⁻¹(x)) = x holds true 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 that applying a function and then its inverse will return the original input, which is crucial for evaluating expressions involving inverse trigonometric functions.

Trigonometric Functions and Their Inverses

Trigonometric functions such as secant (sec) and their inverses, like sec⁻¹, are fundamental in trigonometry. The secant function is defined as the reciprocal of the cosine function, sec(x) = 1/cos(x). Understanding how to manipulate these functions and their inverses is essential for solving problems that involve finding exact values of trigonometric expressions.

Domain and Range of Inverse Functions

The domain and range of inverse functions are critical for determining valid inputs and outputs. For example, the domain of sec⁻¹(x) is x ≥ 1 or x ≤ -1, as sec(x) can only take these values. Knowing the restrictions on the domains and ranges of trigonometric and inverse trigonometric functions helps ensure that calculations are valid and that the results are meaningful.