Determine algebraically whether or not the function
f(x)=-x^3-2x^2+5 is even or odd, and justify your answer.
(1 point)
Responses
The function is odd because f(-x) =-f(x)
The function is odd because f(-x) = f(x)
The function is even because f(-x) = -f(x)
The function is even because f(-x) = f(x)
The function is neither even nor odd because f(-x) doesn't equal f(x) and F(-x) doesn't equal -f(x)
The correct response is: The function is neither even nor odd because f(-x) doesn't equal f(x) and f(-x) doesn't equal -f(x)
The correct response is: The function is neither even nor odd because f(-x) doesn't equal f(x) and F(-x) doesn't equal -f(x).
To determine whether a function is even or odd, we need to check if it satisfies the conditions of even or odd functions.
For a function to be even, it must satisfy the condition f(-x) = f(x) for all x in the domain.
For a function to be odd, it must satisfy the condition f(-x) = -f(x) for all x in the domain.
Let's test these conditions for the given function f(x) = -x^3 - 2x^2 + 5:
f(-x) = -(-x)^3 - 2(-x)^2 + 5
= -(-x^3) - 2x^2 + 5
= x^3 - 2x^2 + 5
From this equation, we can see that f(-x) is not equal to f(x). Thus, the function is not even.
Now, let's test the condition for odd:
-f(x) = -(-x^3 - 2x^2 + 5)
= x^3 + 2x^2 - 5
From this equation, we can see that -f(x) is not equal to f(-x). Thus, the function is not odd either.
Therefore, the function is neither even nor odd.
To determine whether the function f(x) = -x^3 - 2x^2 + 5 is even or odd, we need to evaluate whether f(-x) is equal to f(x) or -f(x).
1. Substitute -x for x in the function: f(-x) = -(-x)^3 - 2(-x)^2 + 5.
Simplify this expression: f(-x) = -(-x^3) - 2(x^2) + 5.
Simplify further: f(-x) = x^3 - 2x^2 + 5.
2. Now, compare this with the original function f(x) = -x^3 - 2x^2 + 5.
We can observe that f(-x) is not equal to f(x).
3. Since f(-x) is not equal to f(x), we now check if f(-x) is equal to -f(x).
Substitute -x for x again in the original function: -f(x) = -(-x^3 - 2x^2 + 5)
Simplify: -f(x) = x^3 + 2x^2 - 5.
4. Comparing f(-x) = x^3 - 2x^2 + 5 with -f(x) = x^3 + 2x^2 - 5, we can see that f(-x) is not equal to -f(x) either.
5. Since f(-x) is not equal to f(x) and f(-x) is not equal to -f(x), we can conclude that the function f(x) = -x^3 - 2x^2 + 5 is neither even nor odd.
Therefore, the correct answer is: "The function is neither even nor odd because f(-x) doesn't equal f(x) and f(-x) doesn't equal -f(x)."