determine algebraically wether or not the function f(x)=-x^3-2x^2+5 is even or odd and justify your answer
A the fumction is odd because f(-x)=-f(x)
Bthe function is odd becauase f(-x)=f(x)
Cthe function is even because f(-x0=-f(x)
D the function is even because f(-x)=f(x)
E the function is neither even nor odd because
F(-x)=f(x) and (-x)=-f(x)
The function f(x) = -x^3 - 2x^2 + 5 is odd because f(-x) = -(-x)^3 - 2(-x)^2 + 5 = -x^3 - 2x^2 + 5 = -f(x).
Therefore, the correct answer is A) the function is odd because f(-x) = -f(x).
To determine whether the function f(x) = -x^3 - 2x^2 + 5 is even or odd, we need to check if it satisfies the properties of even and odd functions.
An even function satisfies the condition f(-x) = f(x) for all values of x.
An odd function satisfies the condition f(-x) = -f(x) for all values of x.
Let's check these conditions for the given function:
1. Check for evenness - f(-x) = f(x)
-x is substituted for x in the function:
f(-x) = -(-x)^3 - 2(-x)^2 + 5
= -(-x^3) - 2x^2 + 5
= x^3 - 2x^2 + 5
This is not equal to the original function, f(x) = -x^3 - 2x^2 + 5.
2. Check for oddness - f(-x) = -f(x)
-x is substituted for x in the function:
f(-x) = -(-x)^3 - 2(-x)^2 + 5
= -(-x^3) - 2x^2 + 5
= -x^3 + 2x^2 + 5
This is not equal to the negation of the original function, -f(x) = -(-x^3 - 2x^2 + 5) = x^3 + 2x^2 - 5.
Since the function does not satisfy either the condition for evenness or oddness, we can conclude that the function f(x) = -x^3 - 2x^2 + 5 is neither even nor odd (Option E).
To determine whether a function is even or odd algebraically, we need to evaluate the function at both f(x) and f(-x) and compare the results.
Let's start with the function f(x) = -x^3 - 2x^2 + 5.
1. To check if the function is even, we need to evaluate f(x) and f(-x) and see if they are equal.
Let's evaluate f(x):
f(x) = -x^3 - 2x^2 + 5
Now, let's evaluate f(-x):
f(-x) = -(-x)^3 - 2(-x)^2 + 5
= -(-x^3) - 2(x^2) + 5
= x^3 - 2x^2 + 5
Since f(x) = -x^3 - 2x^2 + 5 and f(-x) = x^3 - 2x^2 + 5, we can see that f(x) and f(-x) are NOT equal.
Therefore, we can conclude that the function f(x) = -x^3 - 2x^2 + 5 is NOT even.
2. To check if the function is odd, we need to evaluate f(x) and f(-x) and see if they have opposite signs.
Let's evaluate f(x):
f(x) = -x^3 - 2x^2 + 5
Now, let's evaluate f(-x):
f(-x) = -(-x)^3 - 2(-x)^2 + 5
= -(-x^3) - 2(x^2) + 5
= x^3 - 2x^2 + 5
Notice that f(x) = -x^3 - 2x^2 + 5 and f(-x) = x^3 - 2x^2 + 5 have the same signs for all terms.
Therefore, we can conclude that the function f(x) = -x^3 - 2x^2 + 5 is NOT odd.
So, the correct answer is E. The function is neither even nor odd because f(-x) is not equal to f(x) and f(x) does not have opposite signs when substituted with -x.