What is the range of (2x^2+3x-2)/(x-2) ?

The range of (2x^2+3x-2)/(x-2) is f(x) = 4sqrt(6)+11

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range of (2x^2+3x-2)/(x-2)

range

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

f (x) \leq - 46 + 11 or f (x) \geq 46 + 11

Interval Notation

(- \infty, - 46 + 11] \cup [46 + 11, \infty)

Solution steps

Rewrite as

\frac{2 x ^{2} + 3 x - 2}{x - 2} = y

Multiply both sides by

x - 2

2 x^{2} + 3 x - 2 = y (x - 2)

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

Discriminant

2 x^{2} + 3 x - 2 = y (x - 2) : y^{2} - 22 y + 25

y^{2} - 22 y + 25 \geq 0 : y \leq - 46 + 11 or y \geq 46 + 11

Check if the range interval endpoints are included

y = - 46 + 11 :

Included

y = 46 + 11 :

Included

Therefore the range is

f (x) \leq - 46 + 11 or f (x) \geq 46 + 11

\frac{x ^{2} - 81}{x ( x - 9 )}

(4, 6), (- 3, - 7)

f (x) = - 2 x^{2} - 2 x + 30

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

\frac{2 ^{x} - 8}{4 - 2 ^{x}}

What is the range of (2x^2+3x-2)/(x-2) ?
The range of (2x^2+3x-2)/(x-2) is f(x)<=-4sqrt(6)+11\lor f(x)>= 4sqrt(6)+11