What is the range of (x^2+x)/(-2x^2-2x+12) ?

The range of (x^2+x)/(-2x^2-2x+12) is f(x) =-1/50

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range of (x^2+x)/(-2x^2-2x+12)

range

\frac{x ^{2} + x}{- 2 x ^{2} - 2 x + 12}

f (x) < - \frac{1}{2} or f (x) \geq - \frac{1}{50}

Interval Notation

(- \infty, - \frac{1}{2}) \cup [- \frac{1}{50}, \infty)

Solution steps

Rewrite as

\frac{x ^{2} + x}{- 2 x ^{2} - 2 x + 12} = y

Multiply both sides by

- 2 x^{2} - 2 x + 12

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

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

Discriminant

x^{2} + x = y (- 2 x^{2} - 2 x + 12) : 100 y^{2} + 52 y + 1

100 y^{2} + 52 y + 1 \geq 0 : y \leq - \frac{1}{2} or y \geq - \frac{1}{50}

Check if the range interval endpoints are included

y = - \frac{1}{2} :

Excluded

y = - \frac{1}{50} :

Included

Therefore the range is

f (x) < - \frac{1}{2} or f (x) \geq - \frac{1}{50}

What is the range of (x^2+x)/(-2x^2-2x+12) ?
The range of (x^2+x)/(-2x^2-2x+12) is f(x)<-1/2 \lor f(x)>=-1/50