What is the range of (1+x^2)/(x^2) ?

The range of (1+x^2)/(x^2) is f(x)>1

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range of (1+x^2)/(x^2)

range

\frac{1 + x ^{2}}{x ^{2}}

f (x) > 1

Solution steps

Rewrite as

\frac{1 + x ^{2}}{x ^{2}} = y

Multiply both sides by

x^{2}

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

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

Discriminant

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

4 y - 4 \geq 0 : y \geq 1

Check if the range interval endpoints are included

y = 1 :

Excluded

Therefore the range is

f (x) > 1

\frac{1}{arccos ( t - 2 )}

f (x) = (x - 1)^{2} - 9

(5.1) (4.2)

f (x) = \frac{x + 12}{x - 6}

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