To determine whether the given function, f(x)=(-x^3)/(5x^2 - 4) is even, odd, or neither, we will use the properties of even and odd functions. An even function is one for which f(-x) = f(x) for all x in the function's domain. An odd function is a function for which f(-x) = -f(x) for all x in the function's domain.
So first, let's determine f(-x) and compare it to f(x) and -f(x).
To find f(-x):
Take the function f(x) = (-x^3)/(5x^2 - 4) and replace every x with -x to find f(-x):
f(-x) = -(-x)^3 / (5(-x)^2 - 4)
This simplifies to:
= -(-x^3) / (5x^2 -4)
= x^3 / (5x^2 - 4)
Now we compare this to the original function, f(x), and to -f(x):
f(x) = (-x^3) / (5x^2 - 4)
-f(x) = -[(-x^3) / (5x^2 - 4)] = x^3 / (5x^2 - 4)
From these comparisons, we can see that:
f(-x) ≠ f(x), since this would imply that the function is even.
f(-x) = -f(x), which is the condition for a function to be odd.
Therefore, the function f(x) = (-x^3)/(5x^2 - 4) is an odd function.