Algebraically, determine whether the following function is even, odd, or neither. Make sure to show your work
f(x)= x^3 - 2x + 3
A. Neither
B. Even
C. Odd
To determine whether the function is even, odd, or neither, we need to test if the function satisfies the following conditions:
1. If f(-x) = f(x), then the function is even.
2. If f(-x) = -f(x), then the function is odd.
Let's test these conditions for the function f(x) = x^3 - 2x + 3:
1. f(-x) = (-x)^3 - 2(-x) + 3 = -x^3 + 2x + 3
f(x) = x^3 - 2x + 3
Comparing f(-x) and f(x), we can see that f(-x) is not equal to f(x). Therefore, the function does not satisfy the condition for being even.
2. f(-x) = (-x)^3 - 2(-x) + 3 = -x^3 + 2x + 3
-f(x) = -(x^3 - 2x + 3) = -x^3 + 2x - 3
Comparing f(-x) and -f(x), we can also see that f(-x) is not equal to -f(x). Therefore, the function does not satisfy the condition for being odd.
Since the function does not satisfy the conditions for being even or odd, the function is neither even nor odd.
Therefore, the answer is:
A. Neither