Determine algebraically whether or not the function f(×)=-×^3-2×^2+5 is even or odd, and justify your answer.
a. The function is odd because f(-×)=-f(×)
b. The function is odd because f(-×)=f(×)
c. The function is even because f(-×)=-f(×)
d. The function is even because f(-×)=f(×)
e. The function is neither even nor odd because f (-×)=f (×) and f (-×) = -f (×)
To determine whether the function f(×)=-×^3-2×^2+5 is even or odd algebraically, we need to substitute -× for × and compare the result with the original function.
1. Substitute -× into the function: f(-×) = -(-×)^3 - 2(-×)^2 + 5
Simplify: f(-×) = ×^3 - 2×^2 + 5
2. Compare f(-×) with the original function f(×):
- If f(-×) = f(×), the function is even.
- If f(-×) = -f(×), the function is odd.
Comparing f(-×) = ×^3 - 2×^2 + 5 with f(×) = -×^3 - 2×^2 + 5, we can see that they are NOT equal, and f(-×) ≠ -f(×).
Therefore, the function f(×)=-×^3-2×^2+5 is neither even nor odd.
The correct answer is e. The function is neither even nor odd because f(-×) = f(×) and f(-×) ≠ -f(×).
The correct answer is c. The function is even because f(-×)=-f(×).
To determine whether a function is even or odd, we need to analyze the function's symmetry behavior.
A function is even if it satisfies the condition f(-×)=f(×), meaning that if we substitute -× into the function, we get the same result as when we substitute × into the function. In this case, f(-×)= -(-×)^3 -2(-×)^2 + 5 = -(-×^3) -2×^2 + 5 = ×^3 - 2×^2 + 5.
A function is odd if it satisfies the condition f(-×)=-f(×), meaning that if we substitute -× into the function, we get the negative of the result when we substitute × into the function. However, in this case, f(-×) is not equal to -f(×).
Since the function satisfies the condition f(-×)=-f(×), we can conclude that the function is even.
To determine whether a function is even or odd, we need to analyze the behavior of the function when we substitute -x for x.
In this case, we have the function f(x) = -x^3 - 2x^2 + 5.
To check if the function is even, we need to determine if f(-x) is equal to f(x):
f(-x) = -(-x)^3 - 2(-x)^2 + 5
= -(-x^3) - 2(x^2) + 5
= x^3 - 2x^2 + 5
On the other hand, to check if the function is odd, we need to determine if f(-x) is equal to -f(x):
-f(x) = -(-x^3 - 2x^2 + 5)
= x^3 + 2x^2 - 5
Comparing the two expressions, we can see that f(-x) is equal to f(x), but it is not equal to -f(x).
Therefore, the correct answer is option b. The function is odd because f(-x) = f(x).