Question

Which of the following could be found in a table of values for the function g(x)=cos(−1.5x)−1 ?(1 point) Responses (2π/3,0) l, There is no viable option provided. . (2π3,−1), (2π3,−2)

Answer 1

To determine which of the given points could be found in the table of values for the function \( g(x) = \cos(-1.5x) - 1 \), we can evaluate the function at \( x = \frac{2\pi}{3} \) and check the output.

1. Calculate \( g\left(\frac{2\pi}{3}\right) \):
\[
g\left(\frac{2\pi}{3}\right) = \cos\left(-1.5 \cdot \frac{2\pi}{3}\right) - 1
\]
Simplifying the argument:
\[
-1.5 \cdot \frac{2\pi}{3} = -\pi
\]
Now, evaluate \( \cos(-\pi) \):
\[
\cos(-\pi) = -1
\]
Thus,
\[
g\left(\frac{2\pi}{3}\right) = -1 - 1 = -2
\]

From the calculations, we find:
- \( g\left(\frac{2\pi}{3}\right) = -2 \)

Now we can look at the provided options:
1. \( \left(\frac{2\pi}{3}, 0\right) \)
2. "There is no viable option provided."
3. \( \left(\frac{2\pi}{3}, -1\right) \)
4. \( \left(\frac{2\pi}{3}, -2\right) \)

The only point that matches our result is:
- \( \left(\frac{2\pi}{3}, -2\right) \)

So, the correct response is \( \left(\frac{2\pi}{3}, -2\right) \).