Which Of The Following Are Vertical Asymptotes Of F ( X ) = 3 Sec ⁡ ( 5 X F(x)=3 \sec (5x F ( X ) = 3 Sec ( 5 X ]?A. X = Π 10 X=\frac{\pi}{10} X = 10 Π ​ B. X = Π 5 X=\frac{\pi}{5} X = 5 Π ​ C. X = 3 Π 10 X=\frac{3\pi}{10} X = 10 3 Π ​ D. X = 3 Π 5 X=\frac{3\pi}{5} X = 5 3 Π ​

Introduction

In mathematics, a vertical asymptote is a vertical line that a function approaches but never touches. It is a line that the function gets arbitrarily close to, but never crosses. In the context of trigonometric functions, vertical asymptotes occur when the denominator of the function is equal to zero. In this article, we will explore the concept of vertical asymptotes in trigonometric functions, specifically in the function $f(x)=3 \sec (5x)$.

Understanding the Function

The function $f(x)=3 \sec (5x)$ is a trigonometric function that involves the secant function. The secant function is defined as the reciprocal of the cosine function, i.e., $\sec(x) = \frac{1}{\cos(x)}$. The function $f(x)$ is a multiple of the secant function, where the multiple is 3.

Vertical Asymptotes of the Secant Function

The secant function has vertical asymptotes at points where the cosine function is equal to zero. This is because the secant function is the reciprocal of the cosine function, and division by zero is undefined. The cosine function is equal to zero at odd multiples of $\frac{\pi}{2}$, i.e., $\cos(x) = 0$ when $x = \frac{(2n+1)\pi}{2}$, where $n$ is an integer.

Finding Vertical Asymptotes of $f(x)=3 \sec (5x)$

To find the vertical asymptotes of the function $f(x)=3 \sec (5x)$, we need to find the values of $x$ that make the denominator of the function equal to zero. In this case, the denominator is the cosine function, which is equal to zero at odd multiples of $\frac{\pi}{2}$. However, the function $f(x)$ involves the secant function, which is a multiple of the cosine function. Therefore, we need to find the values of $x$ that make the cosine function equal to zero, and then multiply those values by 5.

Calculating Vertical Asymptotes

To calculate the vertical asymptotes of the function $f(x)=3 \sec (5x)$, we need to find the values of $x$ that make the cosine function equal to zero. We can do this by setting the cosine function equal to zero and solving for $x$.

$$\cos(5x) = 0$$

We can use the identity $\cos(x) = 0$ when $x = \frac{(2n+1)\pi}{2}$ to solve for $x$.

$$5x = \frac{(2n+1)\pi}{2}$$

Dividing both sides by 5, we get:

$$x = \frac{(2n+1)\pi}{10}$$

This is the general formula for the vertical asymptotes of the function $f(x)=3 \sec (5x)$.

Evaluating Answer Choices

Now that we have the general formula for the vertical asymptotes of the function $f(x)=3 \sec (5x)$, we can evaluate the answer choices.

A. $x=\frac{\pi}{10}$

This value of $x$ does not satisfy the general formula for the vertical asymptotes of the function $f(x)=3 \sec (5x)$.

B. $x=\frac{\pi}{5}$

This value of $x$ does not satisfy the general formula for the vertical asymptotes of the function $f(x)=3 \sec (5x)$.

C. $x=\frac{3\pi}{10}$

This value of $x$ satisfies the general formula for the vertical asymptotes of the function $f(x)=3 \sec (5x)$.

D. $x=\frac{3\pi}{5}$

This value of $x$ does not satisfy the general formula for the vertical asymptotes of the function $f(x)=3 \sec (5x)$.

Conclusion

In conclusion, the vertical asymptotes of the function $f(x)=3 \sec (5x)$ are given by the general formula $x = \frac{(2n+1)\pi}{10}$, where $n$ is an integer. The correct answer choices are the values of $x$ that satisfy this formula. Therefore, the correct answer is:

• C. $x=\frac{3\pi}{10}$

This is the only value of $x$ that satisfies the general formula for the vertical asymptotes of the function $f(x)=3 \sec (5x)$.

Final Answer

Introduction

In our previous article, we discussed the concept of vertical asymptotes in trigonometric functions, specifically in the function $f(x)=3 \sec (5x)$. We also derived the general formula for the vertical asymptotes of this function. In this article, we will provide a comprehensive Q&A section to help you better understand the concept of vertical asymptotes in trigonometric functions.

Q&A Section

Q1: What is a vertical asymptote?

A1: A vertical asymptote is a vertical line that a function approaches but never touches. It is a line that the function gets arbitrarily close to, but never crosses.

Q2: When do vertical asymptotes occur in trigonometric functions?

A2: Vertical asymptotes occur in trigonometric functions when the denominator of the function is equal to zero. This is because division by zero is undefined.

Q3: What is the general formula for the vertical asymptotes of the function $f(x)=3 \sec (5x)$?

A3: The general formula for the vertical asymptotes of the function $f(x)=3 \sec (5x)$ is given by $x = \frac{(2n+1)\pi}{10}$, where $n$ is an integer.

Q4: How do I find the vertical asymptotes of a trigonometric function?

A4: To find the vertical asymptotes of a trigonometric function, you need to find the values of $x$ that make the denominator of the function equal to zero. You can do this by setting the denominator equal to zero and solving for $x$.

Q5: What is the difference between a vertical asymptote and a hole in a graph?

A5: A vertical asymptote is a vertical line that a function approaches but never touches, while a hole in a graph is a point where the function is not defined. However, the function may still approach the hole as $x$ gets arbitrarily close to it.

Q6: Can a function have multiple vertical asymptotes?

A6: Yes, a function can have multiple vertical asymptotes. This occurs when the denominator of the function is equal to zero at multiple points.

Q7: How do I determine if a function has a vertical asymptote at a particular point?

A7: To determine if a function has a vertical asymptote at a particular point, you need to check if the denominator of the function is equal to zero at that point. If it is, then the function has a vertical asymptote at that point.

Q8: Can a function have a vertical asymptote at a point where the numerator is also equal to zero?

A8: No, a function cannot have a vertical asymptote at a point where the numerator is also equal to zero. This is because the function would be undefined at that point, and would not approach it as $x$ gets arbitrarily close to it.

Q9: How do I graph a function with vertical asymptotes?

A9: To graph a function with vertical asymptotes, you need to plot the function and then draw vertical lines at the points where the function has vertical asymptotes. The function will approach these lines as $x$ gets arbitrarily close to them.

Q10: Can a function have a vertical asymptote at a point where the function is not defined?

A10: No, a function cannot have a vertical asymptote at a point where the function is not defined. This is because the function would not approach that point as $x$ gets arbitrarily close to it.

Conclusion

In conclusion, the concept of vertical asymptotes in trigonometric functions is an important one to understand. By following the general formula for the vertical asymptotes of a function, you can determine the points where the function has vertical asymptotes. We hope that this Q&A section has helped you better understand the concept of vertical asymptotes in trigonometric functions.

Final Answer

The final answer is C.