Which Of The Following Are Vertical Asymptotes Of The Function Y = 2 Cot ⁡ ( 3 X ) + 4 Y=2 \cot (3x)+4 Y = 2 Cot ( 3 X ) + 4 ? Check All That Apply.A. X = 0 X=0 X = 0 B. X = ± Π 2 X=\pm \frac{\pi}{2} X = ± 2 Π ​ C. X = 2 Π X=2\pi X = 2 Π D. X = Π 3 X=\frac{\pi}{3} X = 3 Π ​

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Vertical Asymptotes of the Function y=2cot(3x)+4y=2 \cot (3x)+4

Understanding Vertical Asymptotes

Vertical asymptotes are vertical lines that a function approaches but never touches. They occur when the function is undefined at a particular point, often due to division by zero or a denominator that equals zero. In the case of trigonometric functions, vertical asymptotes typically occur when the denominator of the function equals zero.

The Function y=2cot(3x)+4y=2 \cot (3x)+4

The given function is y=2cot(3x)+4y=2 \cot (3x)+4. To find the vertical asymptotes, we need to determine when the function is undefined. The cotangent function, cot(x)\cot(x), is undefined when its denominator equals zero. In this case, the denominator is cos(3x)\cos(3x).

Finding Vertical Asymptotes

To find the vertical asymptotes, we need to solve the equation cos(3x)=0\cos(3x) = 0. This is because the cotangent function is undefined when its denominator equals zero.

The cosine function equals zero at odd multiples of π2\frac{\pi}{2}. Therefore, we can write the equation as:

3x=(2n+1)π23x = (2n+1)\frac{\pi}{2}

where nn is an integer.

Solving for xx, we get:

x=(2n+1)π6x = \frac{(2n+1)\pi}{6}

This is the general form of the vertical asymptotes. However, we need to check if any of the given options match this form.

Checking the Options

Let's check each of the given options to see if they match the form of the vertical asymptotes.

A. x=0x=0

This option does not match the form of the vertical asymptotes. We can plug in n=0n=0 to get x=π6x=\frac{\pi}{6}, but this is not equal to zero.

B. x=±π2x=\pm \frac{\pi}{2}

This option does not match the form of the vertical asymptotes. We can plug in n=0n=0 to get x=π6x=\frac{\pi}{6}, but this is not equal to ±π2\pm \frac{\pi}{2}.

C. x=2πx=2\pi

This option does not match the form of the vertical asymptotes. We can plug in n=2n=2 to get x=5π6x=\frac{5\pi}{6}, but this is not equal to 2π2\pi.

D. x=π3x=\frac{\pi}{3}

This option does not match the form of the vertical asymptotes. We can plug in n=0n=0 to get x=π6x=\frac{\pi}{6}, but this is not equal to π3\frac{\pi}{3}.

However, we can plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2}, and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6}. But we can also plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get $x
Vertical Asymptotes of the Function y=2cot(3x)+4y=2 \cot (3x)+4 - Q&A

Q: What are vertical asymptotes?

A: Vertical asymptotes are vertical lines that a function approaches but never touches. They occur when the function is undefined at a particular point, often due to division by zero or a denominator that equals zero.

Q: How do you find vertical asymptotes?

A: To find vertical asymptotes, you need to determine when the function is undefined. For trigonometric functions, this typically occurs when the denominator equals zero.

Q: What is the denominator of the function y=2cot(3x)+4y=2 \cot (3x)+4?

A: The denominator of the function y=2cot(3x)+4y=2 \cot (3x)+4 is cos(3x)\cos(3x).

Q: When does the denominator cos(3x)\cos(3x) equal zero?

A: The cosine function equals zero at odd multiples of π2\frac{\pi}{2}. Therefore, we can write the equation as:

3x=(2n+1)π23x = (2n+1)\frac{\pi}{2}

where nn is an integer.

Q: How do you solve the equation 3x=(2n+1)π23x = (2n+1)\frac{\pi}{2}?

A: Solving for xx, we get:

x=(2n+1)π6x = \frac{(2n+1)\pi}{6}

This is the general form of the vertical asymptotes.

Q: Which of the given options match the form of the vertical asymptotes?

A: Let's check each of the given options to see if they match the form of the vertical asymptotes.

A. x=0x=0

This option does not match the form of the vertical asymptotes.

B. x=±π2x=\pm \frac{\pi}{2}

This option does not match the form of the vertical asymptotes.

C. x=2πx=2\pi

This option does not match the form of the vertical asymptotes.

D. x=π3x=\frac{\pi}{3}

This option does not match the form of the vertical asymptotes.

However, we can plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2}, and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6}. But we can also plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get x=3π6=π2x=\frac{3\pi}{6}=\frac{\pi}{2} and then plug in n=1n=-1 to get x=π6x=\frac{\pi}{6} and then plug in n=0n=0 to get x=π6x=\frac{\pi}{6} and then plug in n=1n=1 to get