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

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

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

range

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

f (x) < 1

Interval Notation

(- \infty, 1)

Solution steps

Rewrite as

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

Multiply both sides by

x^{2}

x^{2} - 4 = y x^{2}

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

Discriminant

x^{2} - 4 = y x^{2} : - 16 y + 16

- 16 y + 16 \geq 0 : y \leq 1

Check if the range interval endpoints are included

y = 1 :

Excluded

Therefore the range is

f (x) < 1

\frac{( 5 x + 4 ) ( 4 x - 2 )}{( x ^{2} - 36 ) ^{2}}

2 x + \frac{2}{3} y = 10

- \frac{3}{2 t ^{\frac{3}{2}}}

f (x) = x - 4 + 3

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