Question

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

Are you sure

Answer 2

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 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!