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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 its graph.

Short Answer

Expert verified
The inverse secant function is defined for \(x \leq -1 \) or \(x \geq 1\), and its graph will have vertical asymptotes at \(x = -1\) and \(x = 1\), and horizontal asymptotes at \(y = 0\) and \(y = \pi\).

Step by step solution

01

Defining the Inverse

The inverse secant function, denoted as \(sec^{-1}(x)\) or \(arcsec(x)\) is the inverse function of the secant function, and can be defined by restricting the domain of the secant function to \( [0, \pi/2) \) and \( (\pi/2, \pi] \). It returns the angle whose secant is \(x.\) It's defined for \(x \leq -1 \) or \(x \geq 1\).
02

Understanding Properties

Secant function is positive in the first and fourth quadrants, and negative in the second and third. Its values are always \( \leq -1\) or \( \geq 1\). Since it's the inverse, \(sec^{-1}(x)\) returns values in \( [0, \pi/2) \) if \(x \geq 1 \), and in \( (\pi/2, \pi] \) if \(x \leq -1 \), as those are the ranges where secant is defined.
03

Sketching the Graph

The graph of the inverse secant function \(y = sec^{-1}(x)\) is derived from the secant function by reflecting it about the line \(y = x\) and restricting the domain. It will have vertical asymptotes at \(x = -1\) and \(x = 1\), and horizontal asymptotes at \(y = 0\) and \(y = \pi\).

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