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Determine if the function is Odd, Even or neither

Answer
Neither
Even
Odd

We are tasked with determining whether the function \( g(x) = 1 + \cos(x) \) is odd, even, or neither.

### To check if a function is:
- **Even**: \( g(x) = g(-x) \) for all \( x \).
- **Odd**: \( g(-x) = -g(x) \) for all \( x \).

Let's check each case:

1. **Check for Even Function**:
   \[
   g(-x) = 1 + \cos(-x)
   \]
   Using the identity \( \cos(-x) = \cos(x) \):
   \[
   g(-x) = 1 + \cos(x) = g(x)
   \]
   Since \( g(-x) = g(x) \), the function is **even**.

2. **Check for Odd Function**:
   \[
   g(-x) = 1 + \cos(-x) = 1 + \cos(x)
   \]
   But, \( -g(x) = -(1 + \cos(x)) = -1 - \cos(x) \), which is not equal to \( g(-x) \).

Thus, the function is **not odd**.

### Conclusion:
Since \( g(-x) = g(x) \), the function is **even**.

Would you like more details or have further questions?

Here are some related questions for practice:
1. What is the definition of an odd function in terms of \( f(x) \) and \( f(-x) \)?
2. How can you graphically determine if a function is even or odd?
3. What are examples of common odd functions?
4. Can a function be both even and odd? If yes, provide an example.
5. How does the symmetry of a trigonometric function like cosine relate to its evenness?

**Tip:** Even functions exhibit symmetry about the y-axis, while odd functions exhibit symmetry about the origin.

Determine if the function is Odd, Even, or neither.

Learn how to determine if the function g(x) = 1 + cos(x) is odd, even, or neither. Explore key concepts in trigonometry, including the properties of even and odd functions, and use step-by-step logic to identify the correct classification.

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