Algebraically, determine whether the following function is even, odd, or neither. Make sure to include your work for full credit.
f(x)=x3−2x+3
(1 point)
Responses
Odd
Odd
Even
Even
Neither
Neither
To determine whether the function f(x) = x^3 - 2x + 3 is even, odd, or neither, we need to test if it satisfies either the properties of even or odd functions.
1. Even function: A function is even if f(x) = f(-x) for all x in the domain.
Let's test this by substituting -x for x in the given function:
f(-x) = (-x)^3 - 2(-x) + 3
= -x^3 + 2x + 3
Comparing this with the original function f(x) = x^3 - 2x + 3, we can see that f(-x) ≠ f(x). Therefore, f(x) = x^3 - 2x + 3 is not an even function.
2. Odd function: A function is odd if f(x) = -f(-x) for all x in the domain.
Let's test this by substituting -x for x in the given function and multiplying it by -1:
-f(-x) = -(-(x)^3 + 2(-x) + 3)
= x^3 - 2x + 3
Comparing this with the original function f(x) = x^3 - 2x + 3, we can see that f(x) = -f(-x). Therefore, f(x) = x^3 - 2x + 3 is an odd function.
Therefore, the correct answer is: Odd.