What is the range of f(x)= 6/(x^2-64) ?

The range of f(x)= 6/(x^2-64) is f(x) 0

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range of f(x)= 6/(x^2-64)

range

f (x) = \frac{6}{x ^{2} - 64}

f (x) \leq - \frac{3}{32} or f (x) > 0

Interval Notation

(- \infty, - \frac{3}{32}] \cup (0, \infty)

Solution steps

Rewrite as

\frac{6}{x ^{2} - 64} = y

Multiply both sides by

x^{2} - 64

6 = y (x^{2} - 64)

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

Discriminant

6 = y (x^{2} - 64) : 256 y^{2} + 24 y

256 y^{2} + 24 y \geq 0 : y \leq - \frac{3}{32} or y \geq 0

Check if the range interval endpoints are included

y = - \frac{3}{32} :

Included

y = 0 :

Excluded

Therefore the range is

f (x) \leq - \frac{3}{32} or f (x) > 0

f (x) = x^{4} - 2 x^{2} + 3

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

4^{x} + 2

f (x) = - 2 x + 4

f (x) = 2 x^{5} + 2

What is the range of f(x)= 6/(x^2-64) ?
The range of f(x)= 6/(x^2-64) is f(x)<=-3/32 \lor f(x)>0