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

The range of (x^2)/((x+2)(x-3)) is f(x) = 24/25

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

range

\frac{x ^{2}}{( x + 2 ) ( x - 3 )}

f (x) \leq 0 or f (x) \geq \frac{24}{25}

Interval Notation

(- \infty, 0] \cup [\frac{24}{25}, \infty)

Solution steps

Rewrite as

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

Multiply both sides by

(x + 2) (x - 3)

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

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

Discriminant

x^{2} = y (x + 2) (x - 3) : 25 y^{2} - 24 y

25 y^{2} - 24 y \geq 0 : y \leq 0 or y \geq \frac{24}{25}

Check if the range interval endpoints are included

y = 0 :

Included

y = \frac{24}{25} :

Included

Therefore the range is

f (x) \leq 0 or f (x) \geq \frac{24}{25}

What is the range of (x^2)/((x+2)(x-3)) ?
The range of (x^2)/((x+2)(x-3)) is f(x)<= 0\lor f(x)>= 24/25