Question

Find the domain and range of the function. $$ h(t)=\cot t $$

Short answer

The domain of the function \(h(t)=\cot t\) is all real numbers except integer multiples of \(\pi\), and the range of the function is the set of all real numbers.

Step-by-step solution

Determining the Domain

The cotangent function is undefined wherever its denominator, which is the sine function, is equal to zero. The sine function is zero at \(t=n\pi\), where \(n\) is an integer. Therefore, the domain of the cotangent function, \(h(t)=\cot t\), is all real numbers except integer multiples of \(\pi\), which can be represented as \(t \in \mathbb{R} - \{ n\pi : n \in \mathbb{Z} \}\).

Determining the Range

The range of a function consists of all possible output values. The cotangent function spans all real numbers. Thus, the range of the function, \(h(t)=\cot t\), is the set of all real numbers, represented as \(h(t) \in \mathbb{R}\).