which of the following could be found in a table of the values for the function g(x) =cod(-1.5x)-1 a. (2pie/3, -2) b. There is no viable option provided c. (2pie/3, -1) d. (2pie/4,0)

Answer 1

To determine which points might be in the table of values for the function \( g(x) = \cos(-1.5x) - 1 \), we first need to evaluate \( g \) at the specified inputs.

Step 1: Calculate \( g(x) \) for \( x = \frac{2\pi}{3} \)

\[ g\left(\frac{2\pi}{3}\right) = \cos\left(-1.5 \times \frac{2\pi}{3}\right) - 1 \]

Calculating \( -1.5 \times \frac{2\pi}{3} \):

\[ -1.5 \times \frac{2\pi}{3} = -\pi \]

Now, we find \( \cos(-\pi) \):

\[ \cos(-\pi) = -1 \]

Therefore,

\[ g\left(\frac{2\pi}{3}\right) = -1 - 1 = -2 \]

This means the point \( \left(\frac{2\pi}{3}, -2\right) \) is valid.

Step 2: Check the other options.

For \( x = \frac{2\pi}{4} = \frac{\pi}{2} \):

\[ g\left(\frac{\pi}{2}\right) = \cos\left(-1.5 \times \frac{\pi}{2}\right) - 1 \]

Calculating \( -1.5 \times \frac{\pi}{2} \):

\[ -1.5 \times \frac{\pi}{2} = -\frac{3\pi}{4} \]

Now, we find \( \cos\left(-\frac{3\pi}{4}\right) \):

\[ \cos\left(-\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \]

Therefore,

\[ g\left(\frac{\pi}{2}\right) = -\frac{\sqrt{2}}{2} - 1 \]

This value is not zero. Thus \( \left(\frac{2\pi}{4}, 0\right) \) is not valid.

Summary of results:

• The point \( \left(\frac{2\pi}{3}, -2\right) \) is valid.
• The value \( g\left(\frac{2\pi}{3}\right) = -2 \).
• The point \( \left(\frac{2\pi}{4}, 0\right) \) is invalid.

Therefore, the answer to the question "which of the following could be found in a table of values" is:

a. \( \left(\frac{2\pi}{3}, -2\right) \).

Answer 2

Are you sure

Answer 3

Let's double-check the calculations for the function \( g(x) = \cos(-1.5x) - 1 \) to confirm the answer.

1. Evaluating \( g\left(\frac{2\pi}{3}\right) \):

\[ g\left(\frac{2\pi}{3}\right) = \cos\left(-1.5 \cdot \frac{2\pi}{3}\right) - 1 \]

Calculating the argument:

\[ -1.5 \cdot \frac{2\pi}{3} = -\pi \]

Now find \( \cos(-\pi) \):

\[ \cos(-\pi) = -1 \]

Now substitute back into the function:

\[ g\left(\frac{2\pi}{3}\right) = -1 - 1 = -2 \]

This confirms that \( \left(\frac{2\pi}{3}, -2\right) \) is indeed valid.

2. Evaluating \( g\left(\frac{2\pi}{4}\right) = g\left(\frac{\pi}{2}\right) \):

\[ g\left(\frac{\pi}{2}\right) = \cos\left(-1.5 \cdot \frac{\pi}{2}\right) - 1 \]

Calculating the argument:

\[ -1.5 \cdot \frac{\pi}{2} = -\frac{3\pi}{4} \]

Now find \( \cos\left(-\frac{3\pi}{4}\right) \):

\[ \cos\left(-\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \]

Now substitute back into the function:

\[ g\left(\frac{\pi}{2}\right) = -\frac{\sqrt{2}}{2} - 1 \]

This value is approximately \(-1.707\), which is not \(0\), so \( \left(\frac{2\pi}{4}, 0\right) \) is indeed not valid.

3. Other options to check:

• Checking \( (2\pi/3, -1) \): We found that \( g\left(\frac{2\pi}{3}\right) = -2 \), so it is not valid.

• There are no options that show a valid output at the point \( \left(\frac{2\pi}{3}, -1\right) \).

• There is also no point associated with \( (2\pi/4, 0) \) because we confirmed that \( g\left(\frac{\pi}{2}\right) \neq 0 \).

Since:

• \( \left(\frac{2\pi}{3}, -2\right) \) is the only valid point from the options provided,

The correct answer remains:

a. \( (2\pi/3, -2) \)

Let me know if you have other questions or need further assistance!