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

The range of (x^2+2x)/(x-1) is f(x) = 2sqrt(3)+4

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

range

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

f (x) \leq - 23 + 4 or f (x) \geq 23 + 4

Interval Notation

(- \infty, - 23 + 4] \cup [23 + 4, \infty)

Solution steps

Rewrite as

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

Multiply both sides by

x - 1

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

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

Discriminant

x^{2} + 2 x = y (x - 1) : y^{2} - 8 y + 4

y^{2} - 8 y + 4 \geq 0 : y \leq - 23 + 4 or y \geq 23 + 4

Check if the range interval endpoints are included

y = - 23 + 4 :

Included

y = 23 + 4 :

Included

Therefore the range is

f (x) \leq - 23 + 4 or f (x) \geq 23 + 4

y = \frac{1}{3 - x}

y = \frac{1}{2} cos (2 π x)

\frac{2 x ^{2} - 7 x - 15}{x ^{2} - 3 x - 10}

5 x + 2 y = 8

y = \frac{1}{4} x^{2} + x

What is the range of (x^2+2x)/(x-1) ?
The range of (x^2+2x)/(x-1) is f(x)<=-2sqrt(3)+4\lor f(x)>= 2sqrt(3)+4