What is the range of (x^2)/(1-x^2) ?

The range of (x^2)/(1-x^2) is f(x) = 0

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range of (x^2)/(1-x^2)

range

\frac{x ^{2}}{1 - x ^{2}}

f (x) < - 1 or f (x) \geq 0

Interval Notation

(- \infty, - 1) \cup [0, \infty)

Solution steps

Rewrite as

\frac{x ^{2}}{1 - x ^{2}} = y

Multiply both sides by

1 - x^{2}

x^{2} = y (1 - x^{2})

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

Discriminant

x^{2} = y (1 - x^{2}) : 4 y^{2} + 4 y

4 y^{2} + 4 y \geq 0 : y \leq - 1 or y \geq 0

Check if the range interval endpoints are included

y = - 1 :

Excluded

y = 0 :

Included

Therefore the range is

f (x) < - 1 or f (x) \geq 0

f (x) = - 5 x

f (x) = \frac{5 x - 20}{x ^{2} - 4 x}

y = x^{2} - 2 x + 10

3 x - 2 y = 14

\frac{2}{x ^{2} + 1}

What is the range of (x^2)/(1-x^2) ?
The range of (x^2)/(1-x^2) is f(x)<-1\lor f(x)>= 0