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

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

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range of f(x)=(3x^2)/(x^2-4)

range

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

f (x) \leq 0 or f (x) > 3

Interval Notation

(- \infty, 0] \cup (3, \infty)

Solution steps

Rewrite as

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

Multiply both sides by

x^{2} - 4

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

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

Discriminant

3 x^{2} = y (x^{2} - 4) : 16 y^{2} - 48 y

16 y^{2} - 48 y \geq 0 : y \leq 0 or y \geq 3

Check if the range interval endpoints are included

y = 0 :

Included

y = 3 :

Excluded

Therefore the range is

f (x) \leq 0 or f (x) > 3

2 arcsin (\frac{1}{2} x)

f (x) = \frac{e ^{x} - e ^{- x}}{6}

\frac{x + 7}{8 x + 7}

f (x) = - x^{2} + 5

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