Problem
Determine whether the function \(f(x) = x^3 + 2x\) is odd, even, or neither.
Solution
Step 1 ：An even function is a function that satisfies \(f(x) = f(-x)\) for all x in the domain of f.
Step 2 ：An odd function is a function that satisfies \(f(x) = -f(-x)\) for all x in the domain of f.
Step 3 ：To check if the function is even, we compute \(f(-x)\) and compare it to \(f(x)\). \(f(-x) = (-x)^3 + 2(-x) = -x^3 - 2x\), which is not equal to \(f(x)\). So, the function is not even.
Step 4 ：To check if the function is odd, we compute \(-f(x)\) and compare it to \(f(-x)\). \(-f(x) = -(x^3 + 2x) = -x^3 - 2x\), which is equal to \(f(-x)\). So, the function is odd.
