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Chapter 4: Problem 115

Define the inverse cotangent function by restricting the domain of the cotangent function to the interval \((0, \pi)\), and sketch its graph.

Short Answer

Expert verified
The inverse cotangent function, restricted to the interval (0, 蟺), is a decreasing function which is continuous on all real numbers and has horizontal asymptotes at y=0 and y=蟺.

Step by step solution

01

Understand Cotangent Function

First, it's essential to know that the cotangent function, denoted by cot(x), is the reciprocal of the tangent function. Thus, it's defined as cot(x) = cos(x)/sin(x). So, if \(x \in (0, \pi)\), cot(x) is defined for all real numbers except at \(x = \pi/2\).
02

Define the Restricted Cotangent Function

The domain of the cotangent function is restricted to the open interval (0,蟺) to avoid any undefined values. When a function's domain is restricted, it alters the function's behavior within the stated interval. This new function is denoted as \(cot^{-1}(x)\).
03

Understand Characteristics of the Inverse Cotangent Function

The inverse cotangent function \(cot^{-1}(x)\) is a function that provides the value of the angle (between 0 and \(蟺\)) whose cotangent is \(x\). This function is always decreasing and is continuous on all real numbers. It has horizontal asymptotes at \(y = 0\) and \(y = 蟺\).
04

Draw the Graph

To draw the graph of the inverse cotangent function, recall that the inverse of a function is obtained by reflecting the function in the line \(y = x\). However, since it's burdensome to graph cotangent function first and then reflect it, we can use the aforementioned characteristics. Start by drawing the asymptotes at \(y = 0\) and \(y = 蟺\). The function is decreasing, so for negative values of \(x\), the function starts near \(y = 蟺\), and as \(x\) increases, \(y\) decreases, reaching close to 0 but never touching it. Thus, within the interval (0, 蟺), the graph is a curve extending downward from the upper right to the lower left.

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Most popular questions from this chapter

Use a graph to solve the equation on the interval \([-2 \pi, 2 \pi]\). $$ \cot x=1 $$

Evaluate the expression without using a calculator. $$ \tan ^{-1} 0 $$

Use a graphing utility to graph the function. Use the graph to determine the behavior of the function as \(x \rightarrow c\). (a) As \(x \rightarrow 0^{+}\), the value of \(f(x) \rightarrow\) (b) As \(x \rightarrow 0^{-}\), the value of \(f(x) \rightarrow\) (c) As \(x \rightarrow \pi^{+}\), the value of \(f(x) \rightarrow\) (d) As \(x \rightarrow \pi^{-}\), the value of \(f(x) \rightarrow\) $$ f(x)=\cot x $$

Use a graphing utility to graph the function. Include two full periods. $$ y=-\tan 2 x $$

Use a graphing utility to graph the function. Include two full periods. $$ y=\frac{1}{4} \cot \left(x-\frac{\pi}{2}\right) $$
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