determine if the function is even or odd. Make sure to show all written work. secant

To determine if the function \( f(x) = \sec(x) \) is even, odd, or neither, we follow a series of steps. The definitions of even and odd functions are:

- **Even Function:** \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \).
- **Odd Function:** \( f(x) \) is odd if \( f(-x) = -f(x) \) for all \( x \).

### Step 1: Analyze \( f(x) = \sec(x) \)
The secant function is defined as:
\[
f(x) = \sec(x) = \frac{1}{\cos(x)}
\]

### Step 2: Check if the function is even
To check if \( f(x) \) is even, calculate \( f(-x) \) and compare it with \( f(x) \).

\[
f(-x) = \sec(-x) = \frac{1}{\cos(-x)}
\]

Since cosine is an even function (\( \cos(-x) = \cos(x) \)), we have:
\[
f(-x) = \frac{1}{\cos(x)} = \sec(x)
\]

Thus, \( f(-x) = f(x) \), which confirms that \( f(x) \) is an **even function**.

### Conclusion:
The function \( f(x) = \sec(x) \) is **even**.

---

Would you like further clarification or additional details? Here are some related questions:

1. How do you determine if trigonometric functions like sine or tangent are even or odd?
2. Why is cosine an even function?
3. Can a function be neither even nor odd? How do you test for that?
4. What are the graphical implications of a function being even or odd?
5. How does the periodicity of secant affect its evenness?

**Tip:** Even functions are symmetric with respect to the y-axis, while odd functions are symmetric with respect to the origin.

Determine if the function is even or odd. Make sure to show all written work. secant

Understand whether the secant function, sec(x), is even or odd through a comprehensive step-by-step solution. This problem explores key trigonometric concepts, including the properties of even and odd functions and the secant function definition. Suitable for high school students in Grades 10-12.

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