Detemine algebraically whether or not the funtion f(×)=-×^3-2×^2+5 is even or odd, and justify your answer.

Answer 1

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.