What is the range of (x^2+x-2)/(x^2-3x-4) ?

The range of (x^2+x-2)/(x^2-3x-4) is -infinity <f(x)<infinity

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range of (x^2+x-2)/(x^2-3x-4)

range

\frac{x ^{2} + x - 2}{x ^{2} - 3 x - 4}

- \infty < f (x) < \infty

Interval Notation

(- \infty, \infty)

Solution steps

Rewrite as

\frac{x ^{2} + x - 2}{x ^{2} - 3 x - 4} = y

Multiply both sides by

x^{2} - 3 x - 4

x^{2} + x - 2 = y (x^{2} - 3 x - 4)

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

Discriminant

x^{2} + x - 2 = y (x^{2} - 3 x - 4) : 25 y^{2} - 18 y + 9

25 y^{2} - 18 y + 9 \geq 0 :

True for all

y \in R

Therefore the range is

- \infty < f (x) < \infty

What is the range of (x^2+x-2)/(x^2-3x-4) ?
The range of (x^2+x-2)/(x^2-3x-4) is -infinity <f(x)<infinity