What is the range of x/(x^2+x-6) ?

The range of x/(x^2+x-6) is -infinity <f(x)<infinity

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range of x/(x^2+x-6)

range

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

- \infty < f (x) < \infty

Interval Notation

(- \infty, \infty)

Solution steps

Rewrite as

\frac{x}{x ^{2} + x - 6} = y

Multiply both sides by

x^{2} + x - 6

x = y (x^{2} + x - 6)

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

Discriminant

x = y (x^{2} + x - 6) : 25 y^{2} - 2 y + 1

25 y^{2} - 2 y + 1 \geq 0 :

True for all

y \in R

Therefore the range is

- \infty < f (x) < \infty

f (x) = \frac{( 12 x - 3 )}{( 9 x ^{2} - 4 )}

y = 2 x + 12

- 100 x (- 25 x^{2} + 9)

f (x) = \frac{9}{\frac{9}{x}}

\frac{3}{11}, - \frac{3}{11}

What is the range of x/(x^2+x-6) ?
The range of x/(x^2+x-6) is -infinity <f(x)<infinity