What is the range of (x^2)/(x-1) ?

The range of (x^2)/(x-1) is f(x) = 4

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range of (x^2)/(x-1)

range

\frac{x ^{2}}{x - 1}

f (x) \leq 0 or f (x) \geq 4

Interval Notation

(- \infty, 0] \cup [4, \infty)

Solution steps

Rewrite as

\frac{x ^{2}}{x - 1} = y

Multiply both sides by

x - 1

x^{2} = y (x - 1)

The range is the set of y for which the discriminant is greater or equal to zero

Discriminant

x^{2} = y (x - 1) : y^{2} - 4 y

y^{2} - 4 y \geq 0 : y \leq 0 or y \geq 4

Check if the range interval endpoints are included

y = 0 :

Included

y = 4 :

Included

Therefore the range is

f (x) \leq 0 or f (x) \geq 4

f (x) = 3 x e^{8 x}

f (x) = - \frac{9 sin ( x )}{4 x ^{2} - 9 sin ( x )}

2 + 2 x

5^{x - 2} + 4

(x + \frac{2}{3})^{2} - 3

What is the range of (x^2)/(x-1) ?
The range of (x^2)/(x-1) is f(x)<= 0\lor f(x)>= 4