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

The range of ((x+2)^2)/(x-1) is f(x) = 12

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

range

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

f (x) \leq 0 or f (x) \geq 12

Interval Notation

(- \infty, 0] \cup [12, \infty)

Solution steps

Rewrite as

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

Multiply both sides by

x - 1

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

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

Discriminant

(x + 2)^{2} = y (x - 1) : y^{2} - 12 y

y^{2} - 12 y \geq 0 : y \leq 0 or y \geq 12

Check if the range interval endpoints are included

y = 0 :

Included

y = 12 :

Included

Therefore the range is

f (x) \leq 0 or f (x) \geq 12

csc (\frac{π}{3} x + π)

y = lo g_{3} (x)

f (x) = x^{4}

f (x) = - x^{3} + 3 x^{2} + 24 x + 3

f (x) = \frac{- 5 x ^{2}}{( x - 4 ) ( x + 3 )}

What is the range of ((x+2)^2)/(x-1) ?
The range of ((x+2)^2)/(x-1) is f(x)<= 0\lor f(x)>= 12