f(x)=x^2+2 is it a function odd or even or neither?

To determine whether the function \( f(x) = x^2 + 2 \) is odd, even, or neither, we can use the following definitions:

1. **Even function**: A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \).
2. **Odd function**: A function \( f(x) \) is odd if \( f(-x) = -f(x) \) for all \( x \).

Let's check both conditions for \( f(x) = x^2 + 2 \):

1. **Check if \( f(x) \) is even:**
   \[
   f(-x) = (-x)^2 + 2 = x^2 + 2 = f(x)
   \]
   Since \( f(-x) = f(x) \), the function is **even**.

2. **Check if \( f(x) \) is odd:**
   \[
   f(-x) = x^2 + 2
   \]
   The negative of \( f(x) \) would be:
   \[
   -f(x) = -(x^2 + 2) = -x^2 - 2
   \]
   Since \( f(-x) \neq -f(x) \), the function is **not odd**.

### Conclusion:
The function \( f(x) = x^2 + 2 \) is an **even function**.

Would you like more details, or do you have any other questions?

### Related Questions:
1. What are the graphical implications of a function being even or odd?
2. How can we determine the symmetry of a function from its graph?
3. Can a function be both even and odd? Why or why not?
4. What is the geometric interpretation of even and odd functions?
5. How does the parity of a function affect its Fourier series expansion?
6. How do transformations (like translations) affect the parity of a function?
7. Are all quadratic functions even? Explain.
8. How does the concept of even and odd functions apply to polynomials in general?

**Tip:** Remember, even functions are symmetric about the y-axis, while odd functions are symmetric about the origin.

Explore whether the function f(x) = x^2 + 2 is even, odd, or neither with a detailed analysis. Understand the definitions of even and odd functions, their properties, and how to apply them in mathematical contexts. Suitable for high school students in Grades 11-12.

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