What is the range of (x^2-16)/(4x^2) ?

The range of (x^2-16)/(4x^2) is f(x)< 1/4

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range of (x^2-16)/(4x^2)

range

\frac{x ^{2} - 16}{4 x ^{2}}

f (x) < \frac{1}{4}

Interval Notation

(- \infty, \frac{1}{4})

Solution steps

Rewrite as

\frac{x ^{2} - 16}{4 x ^{2}} = y

Multiply both sides by

x^{2}

\frac{x ^{2} - 16}{4} = y x^{2}

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

Discriminant

\frac{x ^{2} - 16}{4} = y x^{2} : - 256 y + 64

- 256 y + 64 \geq 0 : y \leq \frac{1}{4}

Check if the range interval endpoints are included

y = \frac{1}{4} :

Excluded

Therefore the range is

f (x) < \frac{1}{4}

a sy m pt o t es f (x) = \frac{4}{x - 4}

d o main \frac{x}{3 x + 4}

d o main f (x) = e^{- x} - 3

in v erse f (x) = 8 x + 11

cr i t i c a l f (x) = ln (x - 7)

What is the range of (x^2-16)/(4x^2) ?
The range of (x^2-16)/(4x^2) is f(x)< 1/4