What is the range of (2x^2+8x-24)/(x^2+x-12) ?

The range of (2x^2+8x-24)/(x^2+x-12) is -infinity <f(x)<infinity

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range of (2x^2+8x-24)/(x^2+x-12)

range

\frac{2 x ^{2} + 8 x - 24}{x ^{2} + x - 12}

- \infty < f (x) < \infty

Interval Notation

(- \infty, \infty)

Solution steps

Rewrite as

\frac{2 x ^{2} + 8 x - 24}{x ^{2} + x - 12} = y

Multiply both sides by

x^{2} + x - 12

2 x^{2} + 8 x - 24 = y (x^{2} + x - 12)

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

Discriminant

2 x^{2} + 8 x - 24 = y (x^{2} + x - 12) : 49 y^{2} - 208 y + 256

49 y^{2} - 208 y + 256 \geq 0 :

True for all

y \in R

Therefore the range is

- \infty < f (x) < \infty

What is the range of (2x^2+8x-24)/(x^2+x-12) ?
The range of (2x^2+8x-24)/(x^2+x-12) is -infinity <f(x)<infinity