Math Problem Statement

Which of the following functions is an even function?

Solution

An even function is a function f(x)f(x) that satisfies the condition f(x)=f(x)f(-x) = f(x) for all xx 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=x32x2+6y = x^3 - 2x^2 + 6:

    • y(x)=(x)32(x)2+6=x32x2+6y(-x) = (-x)^3 - 2(-x)^2 + 6 = -x^3 - 2x^2 + 6
    • This is not equal to the original function y=x32x2+6y = x^3 - 2x^2 + 6, so this function is not even.
  2. y=x4+2x2y = x^4 + 2x^2:

    • y(x)=(x)4+2(x)2=x4+2x2y(-x) = (-x)^4 + 2(-x)^2 = x^4 + 2x^2
    • This is equal to the original function y=x4+2x2y = x^4 + 2x^2, so this function is even.
  3. y=x42x2+xy = x^4 - 2x^2 + x:

    • y(x)=(x)42(x)2+(x)=x42x2xy(-x) = (-x)^4 - 2(-x)^2 + (-x) = x^4 - 2x^2 - x
    • This is not equal to the original function y=x42x2+xy = x^4 - 2x^2 + x, so this function is not even.
  4. y=x42x3y = x^4 - 2x^3:

    • y(x)=(x)42(x)3=x4+2x3y(-x) = (-x)^4 - 2(-x)^3 = x^4 + 2x^3
    • This is not equal to the original function y=x42x3y = x^4 - 2x^3, so this function is not even.

Conclusion: The correct answer is the second function y=x4+2x2y = x^4 + 2x^2, which is an even function.

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


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 x1x^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.