To find the behavior of the function \( g(x) = \sec(x) \) over the interval \( [0, 2\pi] \), let's break it down:

### 1. Definition of \( \sec(x) \):
The secant function is defined as:
\[
g(x) = \sec(x) = \frac{1}{\cos(x)}
\]
This function is undefined when \( \cos(x) = 0 \), because division by zero is undefined.

### 2. Points where \( \sec(x) \) is undefined:
The cosine function equals zero at the following points in the interval \( [0, 2\pi] \):
\[
x = \frac{\pi}{2}, \, \frac{3\pi}{2}
\]
Thus, \( g(x) = \sec(x) \) has vertical asymptotes at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \).

### 3. Behavior of \( \sec(x) \) between critical points:
- In the interval \( [0, \frac{\pi}{2}) \), \( \cos(x) > 0 \), so \( \sec(x) > 0 \).
- In the interval \( (\frac{\pi}{2}, \pi] \), \( \cos(x) < 0 \), so \( \sec(x) < 0 \).
- In the interval \( [\pi, \frac{3\pi}{2}) \), \( \cos(x) < 0 \), so \( \sec(x) < 0 \).
- In the interval \( (\frac{3\pi}{2}, 2\pi] \), \( \cos(x) > 0 \), so \( \sec(x) > 0 \).

### 4. Range of \( g(x) \):
Since \( \sec(x) \) is the reciprocal of \( \cos(x) \), and \( -1 \leq \cos(x) \leq 1 \), the range of \( \sec(x) \) is:
\[
(-\infty, -1] \cup [1, \infty)
\]
This means that \( \sec(x) \) never takes values between -1 and 1.

### 5. Graphical behavior:
- There are vertical asymptotes at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \).
- The graph alternates between positive and negative values, going to \( +\infty \) near \( x = 0 \), \( \pi \), and \( 2\pi \), and \( -\infty \) near \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \).

Would you like further details on any of these points or a specific graph of the function?

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Here are 5 related questions to expand on this:

1. What is the domain of the secant function?
2. How do you find vertical asymptotes for trigonometric functions?
3. What are the key properties of \( \sec(x) \) and how do they compare with \( \cos(x) \)?
4. How can you determine intervals of increase and decrease for \( \sec(x) \)?
5. How do other reciprocal trigonometric functions behave over \( [0, 2\pi] \)?

**Tip**: The secant function tends to infinity near the points where \( \cos(x) \) approaches zero. Understanding this behavior helps in identifying its vertical asymptotes.

Explore the behavior of the secant function g(x) = sec(x) over the interval [0, 2π], including the identification of vertical asymptotes, intervals of positivity and negativity, and the range of the function. This problem is suitable for high school students in Grades 10-12.

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