Question
ind the vertical asymptotes (if any) of the graph of the function. (Use \( n \) as an arbitrary Integer If necessary. If an answer does not exist, enter O. On \[ f(x)=5 \tan (\pi x) \] \( =\square \) Need Help? Rendit Watchli
Ask by Hamilton Salinas. in the United States
Feb 21,2025
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Answer
The vertical asymptotes of \( f(x) = 5 \tan(\pi x) \) are at \( x = \frac{1}{2} + n \) for any integer \( n \).
Solution
To find the vertical asymptotes of the function \( f(x) = 5 \tan(\pi x) \), we need to determine where the tangent function is undefined. The tangent function has vertical asymptotes at the points where its argument is equal to \( \frac{\pi}{2} + n\pi \), where \( n \) is any integer.
1. **Set the argument of the tangent function equal to the points of discontinuity:**
\[
\pi x = \frac{\pi}{2} + n\pi
\]
2. **Solve for \( x \):**
\[
x = \frac{1}{2} + n
\]
This means that the vertical asymptotes of the function \( f(x) = 5 \tan(\pi x) \) occur at:
\[
x = \frac{1}{2} + n \quad \text{for any integer } n
\]
Thus, the vertical asymptotes are located at \( x = \frac{1}{2}, \frac{3}{2}, \frac{5}{2}, \ldots \) and so on, in both directions.
In conclusion, the vertical asymptotes of the function are:
\[
\boxed{\frac{1}{2} + n}
\] where \( n \) is any integer.
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Simplify this solution Extra Insights
To find the vertical asymptotes of the function \( f(x) = 5 \tan(\pi x) \), we need to identify where the tangent function is undefined, which occurs at odd multiples of \( \frac{\pi}{2} \). This means we set: \[ \pi x = \frac{\pi}{2} + n\pi \] for odd integers \( n \). Simplifying this gives: \[ x = \frac{1}{2} + n \] Thus, the vertical asymptotes occur at \( x = \frac{1}{2} + n \) where \( n \) is any integer. Therefore, the result is: \[ \text{Vertical Asymptotes: } x = \frac{1}{2} + n \text{ (for } n \in \mathbb{Z}\text{)} \]
