What is the range of 3/(x^2-4) ?

The range of 3/(x^2-4) is f(x) 0

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range of 3/(x^2-4)

range

\frac{3}{x ^{2} - 4}

f (x) \leq - \frac{3}{4} or f (x) > 0

Interval Notation

(- \infty, - \frac{3}{4}] \cup (0, \infty)

Solution steps

Rewrite as

\frac{3}{x ^{2} - 4} = y

Multiply both sides by

x^{2} - 4

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

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

Discriminant

3 = y (x^{2} - 4) : 16 y^{2} + 12 y

16 y^{2} + 12 y \geq 0 : y \leq - \frac{3}{4} or y \geq 0

Check if the range interval endpoints are included

y = - \frac{3}{4} :

Included

y = 0 :

Excluded

Therefore the range is

f (x) \leq - \frac{3}{4} or f (x) > 0

f (x) = (x + 3)^{5}

(4, - 5), (8, 7)

f (x) = \frac{7 x - 1}{5 x + 6}

(- 7, 2), (8, 10)

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

What is the range of 3/(x^2-4) ?
The range of 3/(x^2-4) is f(x)<=-3/4 \lor f(x)>0