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

The range of f(x)= 3/((x^2-2x)) is f(x) 0

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

range

f (x) = \frac{3}{( x ^{2} - 2 x )}

f (x) \leq - 3 or f (x) > 0

Interval Notation

(- \infty, - 3] \cup (0, \infty)

Solution steps

Rewrite as

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

Multiply both sides by

x^{2} - 2 x

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

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

Discriminant

3 = y (x^{2} - 2 x) : 4 y^{2} + 12 y

4 y^{2} + 12 y \geq 0 : y \leq - 3 or y \geq 0

Check if the range interval endpoints are included

y = - 3 :

Included

y = 0 :

Excluded

Therefore the range is

f (x) \leq - 3 or f (x) > 0

x^{2} - 2 x + 3

f (x) = 5.5 x + 5.5

f (x) = 1 - cy

f (x) = \frac{x ^{4} - x}{x ^{5} - x}

f (x) = \frac{1 - x}{x + 2}

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