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Which of the following functions is an even function?

To determine which of the given functions is an even function, we need to recall that a function \( f(x) \) is even if it satisfies the condition:

\[
f(-x) = f(x)
\]

Let's evaluate each function:

1. **\( y = x^3 + 2x^2 \):**
   \[
   y(-x) = (-x)^3 + 2(-x)^2 = -x^3 + 2x^2 \neq x^3 + 2x^2
   \]
   This is not an even function.

2. **\( y = \cos x + 2x^2 \):**
   \[
   y(-x) = \cos(-x) + 2(-x)^2 = \cos(x) + 2x^2 = y(x)
   \]
   This is an even function since \( \cos(-x) = \cos(x) \).

3. **\( y = \sin x + 2x^2 \):**
   \[
   y(-x) = \sin(-x) + 2(-x)^2 = -\sin(x) + 2x^2 \neq \sin(x) + 2x^2
   \]
   This is not an even function.

4. **\( y = x^5 + 2x^2 \):**
   \[
   y(-x) = (-x)^5 + 2(-x)^2 = -x^5 + 2x^2 \neq x^5 + 2x^2
   \]
   This is not an even function.

**Conclusion:**
The only even function among the given options is:

\[
y = \cos x + 2x^2
\]

Would you like more details or have any further questions?

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**Related Questions:**

1. What defines an odd function?
2. Can a function be both even and odd? If yes, give an example.
3. How do trigonometric functions like \( \sin x \) and \( \cos x \) behave under the even-odd rule?
4. What is the significance of symmetry in even functions?
5. How do polynomial functions generally behave in terms of evenness or oddness?
6. Can a non-polynomial function be even or odd?
7. What are some practical applications of even and odd functions?
8. How does the evenness of a function affect its graph?

**Tip:** Remember that even functions are symmetric with respect to the y-axis, which is a useful visual check when sketching graphs or solving problems.

Learn how to identify even functions with this detailed explanation. Explore the definition and characteristics of even functions through specific examples like cosine and polynomial functions. Suitable for advanced high school students.

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