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Which of the following functions is an even function?

An even function is a function \( f(x) \) that satisfies the condition \( f(-x) = f(x) \) for all \( x \) in the domain of the function. This means that the function is symmetric with respect to the y-axis.

Let's analyze each of the given functions:

1. \( y = x^3 - 2x^2 + 6 \):
   - \( y(-x) = (-x)^3 - 2(-x)^2 + 6 = -x^3 - 2x^2 + 6 \) 
   - This is not equal to the original function \( y = x^3 - 2x^2 + 6 \), so this function is **not** even.

2. \( y = x^4 + 2x^2 \):
   - \( y(-x) = (-x)^4 + 2(-x)^2 = x^4 + 2x^2 \) 
   - This is equal to the original function \( y = x^4 + 2x^2 \), so this function **is** even.

3. \( y = x^4 - 2x^2 + x \):
   - \( y(-x) = (-x)^4 - 2(-x)^2 + (-x) = x^4 - 2x^2 - x \)
   - This is not equal to the original function \( y = x^4 - 2x^2 + x \), so this function is **not** even.

4. \( y = x^4 - 2x^3 \):
   - \( y(-x) = (-x)^4 - 2(-x)^3 = x^4 + 2x^3 \)
   - This is not equal to the original function \( y = x^4 - 2x^3 \), so this function is **not** even.

**Conclusion:** The correct answer is the second function \( y = x^4 + 2x^2 \), which is an even function.

Would you like more details on this or have any further questions?

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Here are some related questions you might consider:

1. What is the definition of an odd function?
2. Can a function be both even and odd? If so, under what conditions?
3. How do you determine if a polynomial is even or odd?
4. What is the symmetry of odd functions?
5. Can an even function have a non-zero linear term (i.e., a term with \( x^1 \))?
6. What is the general form of an even polynomial function?
7. How do transformations (like translations or reflections) affect the evenness of a function?
8. How can you verify whether a given trigonometric function is even or odd?

**Tip:** When determining whether a function is even, pay close attention to the exponents of each term. If all exponents are even, the function is likely to be even.

Explore the concept of even functions and determine which function among a given set satisfies the criteria. Learn how to identify symmetry in functions with detailed examples and explanations.

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