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

The range of f(x)=((x^2-4))/(3x^2) is f(x)< 1/3

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

range

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

f (x) < \frac{1}{3}

Interval Notation

(- \infty, \frac{1}{3})

Solution steps

Rewrite as

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

Multiply both sides by

x^{2}

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

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

Discriminant

\frac{x ^{2} - 4}{3} = y x^{2} : - 48 y + 16

- 48 y + 16 \geq 0 : y \leq \frac{1}{3}

Check if the range interval endpoints are included

y = \frac{1}{3} :

Excluded

Therefore the range is

f (x) < \frac{1}{3}

\frac{6}{x - 1}

f (x) = (x + 6) 2

3 csc (\frac{x}{2})

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

f (x) = \frac{x - 3}{x ^{2} - x - 6}

What is the range of f(x)=((x^2-4))/(3x^2) ?
The range of f(x)=((x^2-4))/(3x^2) is f(x)< 1/3