What is the range of f(x)= x/((x-2)(x+3)) ?

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

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range of f(x)= x/((x-2)(x+3))

range

f (x) = \frac{x}{( x - 2 ) ( x + 3 )}

- \infty < f (x) < \infty

Interval Notation

(- \infty, \infty)

Solution steps

Rewrite as

\frac{x}{( x - 2 ) ( x + 3 )} = y

Multiply both sides by

(x - 2) (x + 3)

x = y (x - 2) (x + 3)

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

Discriminant

x = y (x - 2) (x + 3) : 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

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