Detemine algebraically whether or not the funtion f(×)=-×^3-2×^2+5 is even or odd, and justify your answer.
To determine if a function is even or odd, we need to evaluate its symmetry under reflections about the y-axis or the origin.
1. Even Function:
A function is even if f(-x) = f(x).
Let's substitute -x for x in the function f(x):
f(-x) = -(-x)^3 - 2(-x)^2 + 5
= -(-x^3) - 2(x^2) + 5
= x^3 - 2x^2 + 5
Now, compare this with the original function f(x):
f(x) = -x^3 - 2x^2 + 5
Since f(-x) = f(x) for every x, the function f(x) is even.
2. Odd Function:
A function is odd if f(-x) = -f(x).
Let's substitute -x for x in the function f(x):
f(-x) = -(-x)^3 - 2(-x)^2 + 5
= -(-x^3) - 2(x^2) + 5
= x^3 - 2x^2 + 5
And, let's negate the original function f(x):
-f(x) = -(-x^3 - 2x^2 + 5)
= x^3 + 2x^2 - 5
Since f(-x) = -f(x) for every x, the function f(x) is odd.
Therefore, the function f(x) = -x^3 - 2x^2 + 5 is both even and odd.
To determine whether a function is even or odd, we need to check the symmetry of the function.
1. Even Function: A function is even if it satisfies the condition f(x) = f(-x) for all x in the domain of the function.
2. Odd Function: A function is odd if it satisfies the condition f(x) = -f(-x) for all x in the domain of the function.
Now, let's determine whether the given function f(x) = -x^3 - 2x^2 + 5 is even or odd algebraically:
1. Even function check:
Replace x with -x in the function: f(-x) = -(-x)^3 - 2(-x)^2 + 5
Simplifying this expression: f(-x) = -x^3 - 2x^2 + 5
Comparing this with the original function f(x) = -x^3 - 2x^2 + 5, we see that f(x) = f(-x). Therefore, the function is even.
2. Odd function check:
Replace x with -x in the function: -f(-x) = -[-(-x)^3 - 2(-x)^2 + 5]
Simplifying this expression: -f(-x) = x^3 + 2x^2 - 5
Comparing this with the original function f(x) = -x^3 - 2x^2 + 5, we observe that -f(-x) ≠ f(x). Therefore, the function is not odd.
In summary, the given function f(x) = -x^3 - 2x^2 + 5 is even.
To determine whether the function f(x) = -x^3 - 2x^2 + 5 is even or odd, we need to analyze its algebraic properties.
A function is considered even if f(x) = f(-x) for all values of x in the domain.
A function is considered odd if f(x) = -f(-x) for all values of x in the domain.
Let's start by checking if the function is even. Substituting -x for x in the function, we get:
f(-x) = -(-x)^3 - 2(-x)^2 + 5
= -(-x^3) - 2(x^2) + 5
= x^3 - 2x^2 + 5
Since the function f(x) = -x^3 - 2x^2 + 5 is not equal to f(-x) = x^3 - 2x^2 + 5, the function is not even.
Next, let's check if the function is odd. Substituting -x for x in the function, we get:
-f(-x) = -(-(x^3) - 2(x^2) + 5)
= -(-x^3 + 2x^2 - 5)
= x^3 - 2x^2 + 5
Comparing this to the original function f(x), we have:
f(x) = -x^3 - 2x^2 + 5
-f(-x) = x^3 - 2x^2 + 5
Since f(x) = -f(-x), the function f(x) = -x^3 - 2x^2 + 5 is odd.
Therefore, the function f(x) = -x^3 - 2x^2 + 5 is odd.