Problem 49 - Precalculus : Building Concepts and Connections

Question

This set of exercises will draw on the ideas presented in this section and your general math background. What are the domain and range of the function \(f(x)=\csc 2 x+3 ?\)

Step-by-Step Solution

Answer

The domain of the function \(f(x) = \csc 2x + 3\) is \(x \ne n\pi / 2\), for all integers \(n\). The range is \(y > 4\) or \(y < 2\).

Step 1 - Identify the period of function

First, observe the function \(\csc 2 x\). The period, \(p\) of the base function \(\csc x\) is \(\pi\), but the coefficient in front of x in \(\csc 2 x\) shows that the period is now \(\pi / 2\) instead.

Step 2 - Determine the domain

The base function is undefined at the points \(x = n\pi\), where \(n\) is an integer. Given the period of our function is \(\pi / 2\), our function will similarly be undefined at the points \(x = n\pi / 2\), with \(n\) being an integer. Therefore, the domain of the function \(f(x) = \csc 2x + 3\) is \(x \ne n\pi / 2\), for all integers \(n\).

Step 3 - Determine the range

For a base cosecant function, the range is \(y > 1\) or \(y < -1\). The addition of the constant term +3 shifts every output of the function up by 3 units, therefore, the range of the given function is \(y > 4\) or \(y < 2\).

Key Concepts

Domain and Range

When discussing trigonometric functions, the domain refers to all the possible input values (x-values) that can produce a valid output for the function. In the case of the cosecant function, it is defined as the reciprocal of the sine function.
For the function \(f(x) = \csc 2x + 3\), the domain is affected by where the sine function is zero because division by zero is undefined.
For \(\csc 2x\), the function is not defined at points where \(\sin 2x = 0\), which occurs at every multiple of \(\pi/2\).

• This results in the function being undefined at \(x = n\pi/2\), where \(n\) is any integer.

This means that for \(f(x) = \csc 2x + 3\), the domain excludes these x-values.
The range of the function identifies all possible output values (y-values) the function can take.
For the base cosecant function, which is the reciprocal of the sine function, the range is such that \(y > 1\) or \(y < -1\). However, adding 3 to \(\csc 2x\) shifts the entire function upward by 3 units.

• Thus, the range of \(f(x) = \csc 2x + 3\) becomes \(y > 4\) or \(y < 2\).

Understanding these shifts in domain and range is crucial for accurately graphing and interpreting the function.

Function Period

The period of a trigonometric function refers to the distance over which the function's graph continuously repeats. For the basic cosecant function \(\csc x\), its period is \(\pi\), meaning the graph repeats every \(\pi\) units.
When we look at \(\csc 2x\) instead, the presence of the coefficient 2 affects its period.

• The period of \(\csc kx\) is calculated as \(\frac{\pi}{k}\).

So, in this case, \(k = 2\), therefore the period of \(\csc 2x\) is calculated to be \(\frac{\pi}{2}\).
This means that the graph of \(\csc 2x\) repeats itself every \(\frac{\pi}{2}\) units along the x-axis.
Knowing the period helps in plotting the function on a graph as we understand how frequently the pattern repeats along the x-axis.

Cosecant Function

The cosecant function, written as \(\csc x\), is one of the fundamental trigonometric functions closely related to the sine function.

• It is defined as the reciprocal of the sine function: \(\csc x = \frac{1}{\sin x}\).

Because of this relationship, the cosecant function inherits characteristics from the sine function, though with some important differences.
One key feature of the cosecant function is that it does not have values within \([-1, 1]\) since it involves division by the sine.

• The primary behavior is that it results in values greater than or equal to 1 or less than or equal to -1 outside of this interval.

The graph of the cosecant function shows repeating patterns with vertical asymptotes at points where the sine function is zero. This is because the reciprocal of zero is undefined.
In the context of transformations, adding constants or adjusting the frequency or phase of the sine (or cosecant) impacts how the graph looks and behaves.
For example, \(\csc 2x + 3\) involves such transformations: the '2' compresses the period, and '+3' vertically shifts the entire curve upwards, thus changing the range.
Understanding these points makes it easier to visualize and work with the cosecant function in various mathematical contexts.