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Chapter 8: Problem 8

True or False The domain of the inverse cotangent function is the set of real numbers.

Short Answer

Expert verified
True

Step by step solution

01

Understanding the inverse cotangent function

The inverse cotangent function, denoted as \(\text{cot}^{-1}(x)\) or \(\text{arccot}(x)\), is the function that returns the angle whose cotangent is \x\.
02

Define the domain of the inverse cotangent function

The domain of the inverse cotangent function \(\text{cot}^{-1}(x)\) is the set of all real numbers. This is because the cotangent function is defined and produces real values for all real numbers.
03

Domain confirmation

We can confirm that the domain of \(\text{cot}^{-1}(x)\) is \(\text{all real numbers}\), since the cotangent function \(\text{cot}(x)\) takes any real number as input and returns a real number, making the inverse applicable to all real values.
04

Answering the question

Based on the information, the statement 'The domain of the inverse cotangent function is the set of real numbers' is true.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

cotangent
The cotangent function, often written as \(\text{cot}(x)\), is a fundamental trigonometric function. It's the reciprocal of the tangent function, meaning \(\text{cot}(x) = \frac{1}{\tan(x)}\). In a right triangle, the cotangent of an angle is the ratio of the adjacent side to the opposite side.
For any angle \(x\), \(\text{cot}(x)\) is defined as long as \(\tan(x)\) is not zero, which means \(x\) cannot be an odd multiple of \(\frac{\pi}{2}\). Otherwise, cotangent provides real values for any real input.
This broad acceptance of real numbers in the input is why it's useful in various mathematical applications.
inverse functions
Inverse functions essentially reverse the effect of the original function. For the cotangent function, the inverse is known as the inverse cotangent or arccotangent, written as \(\text{cot}^{-1}(x)\) or \(\text{arccot}(x)\). This function answers the question: 'For a given value, what angle has this cotangent?'
For example, if \(\text{cot}(x) = y\), then \(\text{cot}^{-1}(y) = x\). The inverse cotangent function maps input values back to their corresponding angles within a specific range, often \(0 \text{ to } \pi\). This characteristic means we can find angles for any cotangent values.
domain
The domain of a function is the set of all possible input values that the function can accept. For the inverse cotangent function \(\text{cot}^{-1}(x)\), the domain is all real numbers. This is because the cotangent function can take any real number as input and output any real number, without restrictions.
As a result, any real value can be input to \(\text{cot}^{-1}(x)\) and it will return a valid angle.
To confirm, remember the definition of the cotangent function: since \(\text{cot}(x)\) is defined for almost all real numbers (excluding odd multiples of \(\frac{\frac{\pi}{2}}\)), its inverse also has a domain that covers all real numbers.

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