Math Problem Statement

Which of the following functions exhibits odd symmetry?

Solution

To determine which function exhibits odd symmetry, we need to recall the condition for odd symmetry in a function f(x)f(x). A function is considered odd if:

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

This means that for every term in the function, the powers of xx must be odd. Let's analyze each option:

  1. f(x)=x5+2x4f(x) = x^5 + 2x^4

    • f(x)=(x)5+2(x)4=x5+2x4f(-x) = (-x)^5 + 2(-x)^4 = -x^5 + 2x^4
    • Since f(x)f(x)f(-x) \neq -f(x), this is not odd.
  2. f(x)=x7x2f(x) = x^7 - x^2

    • f(x)=(x)7(x)2=x7x2f(-x) = (-x)^7 - (-x)^2 = -x^7 - x^2
    • f(x)f(x)f(-x) \neq -f(x), so this is also not odd.
  3. f(x)=x5+3x3f(x) = x^5 + 3x^3

    • f(x)=(x)5+3(x)3=x53x3f(-x) = (-x)^5 + 3(-x)^3 = -x^5 - 3x^3
    • Since f(x)=f(x)f(-x) = -f(x), this function is odd.
  4. f(x)=x3+1f(x) = x^3 + 1

    • f(x)=(x)3+1=x3+1f(-x) = (-x)^3 + 1 = -x^3 + 1
    • f(x)f(x)f(-x) \neq -f(x), so it is not odd.
  5. f(x)=x3+x2f(x) = x^3 + x - 2

    • f(x)=(x)3+(x)2=x3x2f(-x) = (-x)^3 + (-x) - 2 = -x^3 - x - 2
    • f(x)f(x)f(-x) \neq -f(x), so it is not odd.

Thus, the correct answer is:

f(x)=x5+3x3f(x) = x^5 + 3x^3, which exhibits odd symmetry.

Would you like further details or have any questions?

Here are some follow-up questions:

  1. What are the conditions for a function to be even?
  2. How can we verify symmetry in more complex functions?
  3. What is the difference between even and odd symmetry in practical applications?
  4. How do power functions relate to the symmetry of a function?
  5. Can a function exhibit both even and odd symmetry?

Tip: Functions with only odd-powered terms tend to exhibit odd symmetry.

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Math Problem Analysis

Mathematical Concepts

Function Symmetry
Odd Symmetry

Formulas

f(-x) = -f(x)

Theorems

Odd function property

Suitable Grade Level

High School