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.