How can we determine algebraically if the function f(x) = x^5 + x^3 + x is even, odd, or neither?

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To determine whether the function f(x) = x^5 + x^3 + x is even, odd, or neither, we first need to understand the definitions of even and odd functions:

  • A function is considered even if for every x in the domain, f(-x) = f(x).
  • A function is considered odd if for every x in the domain, f(-x) = -f(x).

Now, let’s analyze the function f(x) = x^5 + x^3 + x:

  1. We will first calculate f(-x):
    • f(-x) = (-x)^5 + (-x)^3 + (-x)
    • By applying the properties of exponents, this simplifies to:
    • f(-x) = -x^5 – x^3 – x
  2. Next, we check if f(-x) equals f(x):
    • f(x) = x^5 + x^3 + x
    • Since f(-x) = -x^5 – x^3 – x does not equal f(x), the function is not even.
  3. Now, let’s check if f(-x) equals -f(x):
    • Calculating -f(x):
    • -f(x) = -(x^5 + x^3 + x) = -x^5 – x^3 – x
    • Since f(-x) = -x^5 – x^3 – x equals to -f(x), we conclude the function is odd.

Thus, we can conclude that the function f(x) = x^5 + x^3 + x is an odd function.

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