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

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

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

range

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

f (x) < 2 or f (x) \geq 50

Interval Notation

(- \infty, 2) \cup [50, \infty)

Solution steps

Rewrite as

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

Multiply both sides by

x^{2} + x

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

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

Check if the range interval endpoints are included

y = 2 :

Excluded

y = 50 :

Included

Therefore the range is

f (x) < 2 or f (x) \geq 50

- x^{2} + x - 5

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

\frac{8}{x + 4}

f (x) = \frac{( \frac{5}{x} )}{( \frac{5}{x} ) + 5}

f (x) = x^{2} - 4 x - 12 + \frac{1}{x ^{2}}

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