Question

Finding Vertical Asymptotes In Exercises \(13-28\) , find the vertical asymptotes (if any) of the graph of the function. $$ f(x)=\csc \pi x $$

Short answer

The vertical asymptotes of the function \(f(x) = \csc πx\) are at every integer value of \(x\).

Step-by-step solution

Identify the undefined points of the function

Recall the identity \(\csc \theta = 1/ \sin \theta\). Therefore, the function \(f(x) = \csc πx\) is undefined where the sine function is 0. The sine function is zero at all integer multiples of \(π\). So, \(πx = nπ\) where \(n\) is any integer.

Solve for x

To find the values of \(x\) that will result in a vertical asymptote, solve the equation \(πx = nπ\) for \(x\). This results in \(x = n\).

List all the vertical asymptotes

This means that there are vertical asymptotes at every integer value of \(x\).

Key concepts

Trigonometric Functions

Trigonometric functions deal with angles and relate them to ratios of two sides within right-angled triangles. These functions are periodic, which means they repeat their values in regular intervals. The most common trigonometric functions are sine (\(\sin\theta\)), cosine (\(\cos\theta\)), and tangent (\(\tan\theta\)).

In the exercise provided, \(\csc \pi x\) is the focus. Cosecant, denoted as \(\csc\), is the reciprocal of the sine function. Therefore, \(\csc\theta = \frac{1}{\sin\theta}\).

Understanding cosecant and its relationship with sine is crucial because when sine is zero, cosecant becomes undefined. This leads us to the topic of undefined points and plays a critical role in identifying vertical asymptotes.

Undefined Points

A function is undefined at any point where its expression cannot provide a finite or real value. For the function \(f(x) = \csc \pi x\), this means looking into where \(\sin\theta = 0\).

The sine function equals zero at integer multiples of \(\pi\), such as \(0, \pi, 2\pi, -\pi\), etc. Because cosecant is the reciprocal of sine, the function \(\csc\theta\) is undefined when \(\sin\theta\) is zero.

Thus, for \(\csc \pi x\), it becomes undefined where \(\pi x = n\pi\). Solving this for \(x\), we find \(x = n\), indicating at integer values, the function doesn't produce a valid output. These values of \(x\) are crucial because they guide us to discover the vertical asymptotes.

Solving Equations

I'm sure you've seen equations before that are the keys to solving and answering mathematical problems. Here, the task is to solve \(\pi x = n\pi\) to find where \(f(x) = \csc \pi x\) becomes undefined. Solving this equation involves isolating \(x\):

• Initially, observe \(\pi x = n\pi\)
• Divide both sides by \(\pi\) to isolate \(x\)
• You'll find \(x = n\)

In this step, "\(n\)" represents any integer. This process simplifies identifying not only where the function is undefined but also where vertical asymptotes occur.

Asymptotic Behavior

Asymptotic behavior of a function refers to how it acts as it approaches a particular value, particularly where it may not reach. In this context, vertical asymptotes are lines \(x=a\) where a function goes towards infinity or negative infinity as \(x\) approaches \(a\).

For \(f(x) = \csc \pi x\), the vertical asymptotes appear at \(x = n\), where \(n\) is any integer. Here, as \(x\) nears an integer, the function's value spikes towards infinity because of the reciprocal nature of cosecant relative to the sine.

• These asymptotes indicate boundaries where the function does not reach a finite limit.
• The graph of \(f(x) = \csc \pi x\) will wild oscillations heading toward these vertical lines.

Understanding this behavior helps visualize the function's profile and anticipate its peculiar properties around integer points.