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

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

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range of (2x^2-3)/(x^2)

range

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

f (x) < 2

Interval Notation

(- \infty, 2)

Solution steps

Rewrite as

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

Multiply both sides by

x^{2}

2 x^{2} - 3 = y x^{2}

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

Discriminant

2 x^{2} - 3 = y x^{2} : - 12 y + 24

- 12 y + 24 \geq 0 : y \leq 2

Check if the range interval endpoints are included

y = 2 :

Excluded

Therefore the range is

f (x) < 2