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

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

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

range

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

- \infty < f (x) < \infty

Interval Notation

(- \infty, \infty)

Solution steps

Rewrite as

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

Multiply both sides by

x^{2} - 4

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

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

Discriminant

x^{2} - 3 x - 2 = y (x^{2} - 4) : 16 y^{2} - 24 y + 17

16 y^{2} - 24 y + 17 \geq 0 :

True for all

y \in R

Therefore the range is

- \infty < f (x) < \infty

y = - \frac{6}{7} x

f (x) = \frac{x ^{2} + 8 x - 9}{x ^{2} + 3 x - 4}

x^{3} - 5

\frac{x - 6}{2 x ^{2} - 5 x + 3}

y = 6 x - 8

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