What is the range of f(x)=(3x^2)/(x^2-1) ?

The range of f(x)=(3x^2)/(x^2-1) is f(x) 3

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range of f(x)=(3x^2)/(x^2-1)

range

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

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} - 1} = y

Multiply both sides by

x^{2} - 1

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

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

Discriminant

3 x^{2} = y (x^{2} - 1) : 4 y^{2} - 12 y

4 y^{2} - 12 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

\frac{( x ^{2} + 5 x + 4 )}{x - 2}

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

4 x + 5 y = 20

f (x) = 2 + \frac{6}{5} y

4 x - y = 1

What is the range of f(x)=(3x^2)/(x^2-1) ?
The range of f(x)=(3x^2)/(x^2-1) is f(x)<= 0\lor f(x)>3