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

The range of f(x)=(x^2+x-2)/(x^2) is f(x)<= 9/8

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range of f(x)=(x^2+x-2)/(x^2)

range

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

f (x) \leq \frac{9}{8}

Interval Notation

(- \infty, \frac{9}{8}]

Solution steps

Rewrite as

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

Multiply both sides by

x^{2}

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

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

Discriminant

x^{2} + x - 2 = y x^{2} : - 8 y + 9

- 8 y + 9 \geq 0 : y \leq \frac{9}{8}

Check if the range interval endpoints are included

y = \frac{9}{8} :

Included

Therefore the range is

f (x) \leq \frac{9}{8}

What is the range of f(x)=(x^2+x-2)/(x^2) ?
The range of f(x)=(x^2+x-2)/(x^2) is f(x)<= 9/8