What is the range of f(x)= x/((x-1)(x-2)) ?

The range of f(x)= x/((x-1)(x-2)) is f(x) = 2sqrt(2)-3

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range of f(x)= x/((x-1)(x-2))

range

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

f (x) \leq - 22 - 3 or f (x) \geq 22 - 3

Interval Notation

(- \infty, - 22 - 3] \cup [22 - 3, \infty)

Solution steps

Rewrite as

\frac{x}{( x - 1 ) ( x - 2 )} = y

Multiply both sides by

(x - 1) (x - 2)

x = y (x - 1) (x - 2)

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

Discriminant

x = y (x - 1) (x - 2) : y^{2} + 6 y + 1

y^{2} + 6 y + 1 \geq 0 : y \leq - 22 - 3 or y \geq 22 - 3

Check if the range interval endpoints are included

y = - 22 - 3 :

Included

y = 22 - 3 :

Included

Therefore the range is

f (x) \leq - 22 - 3 or f (x) \geq 22 - 3

y = \frac{x ^{3}}{( x - 1 ) ^{2}}

f (x) = 9 x - 8

4 - (x - 2)^{2}

f (x) = x^{3} - 7 x^{2} + 15 x + 9

f (x) = \frac{x - 4}{x ^{2} + 13 x + 36}

What is the range of f(x)= x/((x-1)(x-2)) ?
The range of f(x)= x/((x-1)(x-2)) is f(x)<=-2sqrt(2)-3\lor f(x)>= 2sqrt(2)-3