To determine if the function f(x) = x^5 + x is odd or even, we need to check if it satisfies the properties of odd and even functions.
An even function is symmetric about the y-axis, which means that f(-x) = f(x) for all x.
A function is odd if it is symmetric about the origin, which means that f(-x) = -f(x) for all x.
Let's check if f(x) satisfies the property of even functions:
f(-x) = (-x)^5 + (-x) = -x^5 - x
f(x) = x^5 + x
f(-x) ≠ f(x)
Since f(-x) ≠ f(x), the function f(x) is not even.
Now let's check if f(x) satisfies the property of odd functions:
f(-x) = (-x)^5 + (-x) = -x^5 - x
-f(x) = -(x^5 + x) = -x^5 - x
f(-x) = -f(x)
Since f(-x) = -f(x), the function f(x) is odd.
Therefore, the function f(x) = x^5 + x is odd.