Which characteristic is correct for the function f(x)=−2x3+3x ?
neither even nor odd
even
both even and odd
odd
The correct characteristic for the function f(x)=−2x^3+3x is "odd".
To determine if a function is even, odd, or neither, we need to understand the definitions of even and odd functions:
1. An even function has symmetry about the y-axis, meaning f(-x) = f(x). In other words, if you reflect the graph across the y-axis, it remains unchanged.
2. An odd function has symmetry about the origin, meaning f(-x) = -f(x). In other words, if you reflect the graph across the origin, it remains unchanged, but is also flipped upside down.
Now, let's apply these definitions to the given function, f(x) = -2x^3 + 3x:
1. For evenness:
- Replace x by -x in the function: f(-x) = -2(-x)^3 + 3(-x) = -2(-x^3) - 3x = 2x^3 - 3x.
- Compare this with the original function, f(x) = -2x^3 + 3x.
- Since f(-x) ≠ f(x), the function is not even.
2. For oddness:
- Replace x by -x in the function: f(-x) = -2(-x)^3 + 3(-x) = -2(-x^3) - 3x = 2x^3 - 3x.
- Compare this with the negation of the original function, -f(x) = -(-2x^3 + 3x) = 2x^3 - 3x.
- Since f(-x) = -f(x), the function is odd.
Therefore, the characteristic that is correct for the function f(x) = -2x^3 + 3x is odd.