Determine the x-intercepts of the function \( g(x) = \cos(x) - 1 \).

To find the **x-intercepts** of the function \( g(x) = \cos(x) - 1 \), we need to determine the values of \( x \) for which \( g(x) = 0 \). ### Step-by-Step Solution: 1. **Set the function equal to zero:** \[ \cos(x) - 1 = 0 \] 2. **Solve for \( \cos(x) \):** \[ \cos(x) = 1 \] 3. **Determine the values of \( x \) where \( \cos(x) = 1 \):** The cosine function equals 1 at all angles that are integer multiples of \( 2\pi \). This is because the cosine function has a period of \( 2\pi \), and it reaches its maximum value of 1 at these points. Therefore: \[ x = 2\pi k \quad \text{where } k \text{ is any integer} \] ### Conclusion: All **x-intercepts** of the function \( g(x) = \cos(x) - 1 \) occur at angles that are integer multiples of \( 2\pi \). **Mathematically, the x-intercepts are:** \[ x = 2\pi k \quad \text{for all integers } k \] ### Example: - \( k = 0 \): \( x = 0 \) - \( k = 1 \): \( x = 2\pi \) - \( k = -1 \): \( x = -2\pi \) - \( k = 2 \): \( x = 4\pi \) - And so on. These are the points where the graph of \( g(x) \) intersects the x-axis. **Graphical Illustration:**  *(Replace the image link with an actual graph if needed.)* **Final Answer:** All real numbers equal to an even multiple of π. In other words, every x with x = 2 π k for some integer k.