Math Problem Statement

Which of the following functions is an even function?

Solution

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

f(x)=f(x)f(-x) = f(x)

Let's evaluate each function:

  1. y=x3+2x2y = x^3 + 2x^2: y(x)=(x)3+2(x)2=x3+2x2x3+2x2y(-x) = (-x)^3 + 2(-x)^2 = -x^3 + 2x^2 \neq x^3 + 2x^2 This is not an even function.

  2. y=cosx+2x2y = \cos x + 2x^2: y(x)=cos(x)+2(x)2=cos(x)+2x2=y(x)y(-x) = \cos(-x) + 2(-x)^2 = \cos(x) + 2x^2 = y(x) This is an even function since cos(x)=cos(x)\cos(-x) = \cos(x).

  3. y=sinx+2x2y = \sin x + 2x^2: y(x)=sin(x)+2(x)2=sin(x)+2x2sin(x)+2x2y(-x) = \sin(-x) + 2(-x)^2 = -\sin(x) + 2x^2 \neq \sin(x) + 2x^2 This is not an even function.

  4. y=x5+2x2y = x^5 + 2x^2: y(x)=(x)5+2(x)2=x5+2x2x5+2x2y(-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=cosx+2x2y = \cos x + 2x^2

Would you like more details or have any further questions?


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 sinx\sin x and cosx\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.

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Math Problem Analysis

Mathematical Concepts

Functions
Even Function

Formulas

Definition of even function: f(-x) = f(x)

Theorems

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Suitable Grade Level

Grades 11-12