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Chapter 4: Problem 116

Define the inverse secant function by restricting the domain of the secant function to the intervals \([0, \pi / 2)\) and \((\pi / 2, \pi],\) and sketch the graph of the inverse trigonometric function.

Short Answer

Expert verified
The inverse secant function, sec\(^{-1}(x)\) can be defined on the intervals \([0, \pi / 2)\) and \((\pi / 2, \pi]\). Its graph is obtained by reflecting the graph of the restricted secant function over the line y = x. The upwards 'U' shaped portion of the secant function turns into a rightward opening 'U' and the downwards 'U' into a leftwards opening 'U' for the inverse secant function.

Step by step solution

01

- Understanding the secant function

We begin by understanding the secant function. The secant function, sec(x), is the reciprocal of the cosine function, meaning sec(x) = 1/cos(x). Notice that sec(x) is undefined when cos(x) = 0.
02

- Restricting the domain

To define an inverse, sec\(^{-1}(x)\), we have to restrict the domain of the secant function so it passes the horizontal line test and becomes a one-to-one function. With the provided intervals, \([0, \pi / 2)\) and \((\pi / 2, \pi]\), we exclude the values where the secant function has vertical asymptotes and is undefined.
03

- Sketching the restricted secant function

On graph paper, sketch the secant function on the restricted intervals. Notice it forms a U shape on the interval \([0, \pi/2)\) and an upside-down U on the interval \((\pi/2, \pi]\). Vertical asymptotes occur at x = \(\pi / 2\).
04

- Sketching the inverse function

The inverse function is the reflection of the original function over the line y = x. So, sketch a line y = x and reflect the restricted secant graph over the line. The upward-opening 'U' shape becomes a rightward-opening 'U' for the inverse secant function, and the downward 'U' becomes a leftward-opening 'U'.

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