Find the range of the function. f(x)=(x-1)^2/x^3-2x^2+x

Range = (-∞,0) U (0,∞)

Step-by-step explanation:

The given function is:

f(x) = (x-1)²/x³-2x²+x

We can reduce the denominator:

= (x-1)²/x(x²-2x+1)

we know that a²-2ab+b² = (a-b)²

= (x-1)²/x(x-1)²

= 1/x

The function 1/x is undefined for x = 0.

Hence, its domain lies in (-∞,0) U (0,∞).

Hence x can either be less than or greater than zero.

The function f(x) is greater than zero when x<0.

The function f(x) is less than zero when x>0.

There is no way the function can be zero because the numerator is constant.

Hence the domain of the function lies in the interval:

(-∞,0) U (0,∞)