Determine whether the following function is odd, even, or neither by setting f(x) to f(-x): `f(x)= x^2/(x-2)^3`

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lemjay | High School Teacher | (Level 2) Senior Educator

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`f(x)= x^2/(x-2)^3`

To determine if the function is even, odd, or neither, replace the x in the function with -x. If the result is f(-x)=f(x), the function is even. If it is f(-x)=-f(x), the function is odd. Otherwise, the function is neither even nor odd.

So, replacing the x with -x in the given function yields,

`f(-x)=(-x)^2/(-x-2)^3`

`f(-x)=x^2/(-(x+2))^3`

`f(-x)=-x^2/(x+2)^3`

Notice that the denominator changes from (x-2)^3 to (x+2)^3. This means that f(-x) is not equal to the original function f(x). It is not also equal to -f(x).

Hence, the given function is neither even nor odd.

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