https://file.aiquickdraw.com/mathbot/file/1727107987201beh95kva.jpg

The given function is \( f(x) = \tan(\pi x) \).

### To find the vertical asymptotes of the function:
1. **Recall the behavior of the tangent function**: The tangent function \( \tan(\theta) \) has vertical asymptotes wherever \( \cos(\theta) = 0 \), which corresponds to \( \theta = \frac{\pi}{2} + n\pi \), where \( n \) is any integer.

2. **Apply this to the function** \( \tan(\pi x) \): We need to find where \( \tan(\pi x) \) becomes undefined, which happens when \( \cos(\pi x) = 0 \).

3. **Solve for \( x \)**:
   \[
   \cos(\pi x) = 0 \quad \text{when} \quad \pi x = \frac{\pi}{2} + n\pi.
   \]
   Divide by \( \pi \):
   \[
   x = \frac{1}{2} + n.
   \]
   Hence, the vertical asymptotes occur at \( x = \frac{1}{2} + n \), where \( n \) is any integer.

### Conclusion:
The vertical asymptotes of the function \( f(x) = \tan(\pi x) \) occur at \( x = \frac{1}{2} + n \), where \( n \) is an integer.

Let me know if you need further details or have any questions!

### Related Questions:
1. What are the horizontal asymptotes of the tangent function?
2. How does the graph of \( \tan(\pi x) \) differ from \( \tan(x) \)?
3. What are the zeros of \( f(x) = \tan(\pi x) \)?
4. How do you determine the period of the function \( \tan(\pi x) \)?
5. Can you explain how to find the vertical asymptotes of \( \sec(\pi x) \)?

### Tip:
For any trigonometric function like tangent, vertical asymptotes occur where the denominator of the associated sine and cosine function becomes zero.

Find the vertical asymptotes (if any) of the graph of the function f(x) = tan(πx). Use n as an arbitrary integer if necessary. If an answer does not exist, enter DNE.

Find the vertical asymptotes of the function f(x) = tan(πx). Understand the behavior of the tangent function and the conditions where it becomes undefined. This solution explains step-by-step how to solve for the vertical asymptotes and provides insights into trigonometric properties of tangent.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation