What is the range of f(x)=(2x^2+2x-4)/(x^2+x) ?

The range of f(x)=(2x^2+2x-4)/(x^2+x) is f(x) = 18

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range of f(x)=(2x^2+2x-4)/(x^2+x)

range

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

f (x) < 2 or f (x) \geq 18

Interval Notation

(- \infty, 2) \cup [18, \infty)

Solution steps

Rewrite as

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

Multiply both sides by

x^{2} + x

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

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

Discriminant

2 x^{2} + 2 x - 4 = y (x^{2} + x) : y^{2} - 20 y + 36

y^{2} - 20 y + 36 \geq 0 : y \leq 2 or y \geq 18

Check if the range interval endpoints are included

y = 2 :

Excluded

y = 18 :

Included

Therefore the range is

f (x) < 2 or f (x) \geq 18

What is the range of f(x)=(2x^2+2x-4)/(x^2+x) ?
The range of f(x)=(2x^2+2x-4)/(x^2+x) is f(x)<2\lor f(x)>= 18