What is the range of (x^2-4)/(7x^2) ?

The range of (x^2-4)/(7x^2) is f(x)< 1/7

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range of (x^2-4)/(7x^2)

range

\frac{x ^{2} - 4}{7 x ^{2}}

f (x) < \frac{1}{7}

Interval Notation

(- \infty, \frac{1}{7})

Solution steps

Rewrite as

\frac{x ^{2} - 4}{7 x ^{2}} = y

Multiply both sides by

x^{2}

\frac{x ^{2} - 4}{7} = 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}{7} = y x^{2} : - 112 y + 16

- 112 y + 16 \geq 0 : y \leq \frac{1}{7}

Check if the range interval endpoints are included

y = \frac{1}{7} :

Excluded

Therefore the range is

f (x) < \frac{1}{7}

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

F (X) = X^{4}

f (x) = lo g_{2} (4 - x^{4})

f (x) = - 4 x

f (x) = - x^{4} - 4 x^{3} + 8 x - 1