What is the range of 1/(4-x^2) ?

The range of 1/(4-x^2) is f(x) = 1/4

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range of 1/(4-x^2)

range

\frac{1}{4 - x ^{2}}

f (x) < 0 or f (x) \geq \frac{1}{4}

Interval Notation

(- \infty, 0) \cup [\frac{1}{4}, \infty)

Solution steps

Rewrite as

\frac{1}{4 - x ^{2}} = y

Multiply both sides by

4 - x^{2}

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

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

Discriminant

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

4 y (4 y - 1) \geq 0 : y \leq 0 or y \geq \frac{1}{4}

Check if the range interval endpoints are included

y = 0 :

Excluded

y = \frac{1}{4} :

Included

Therefore the range is

f (x) < 0 or f (x) \geq \frac{1}{4}

y = x^{3} - 4 x

f (x) = 3 (x + 8)^{7}

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

f (x) = - 2 (\frac{1}{4})^{x - 3}

What is the range of 1/(4-x^2) ?
The range of 1/(4-x^2) is f(x)<0\lor f(x)>= 1/4