What is the range of f(x)=(x^2-6x+12)/(x-4) ?

The range of f(x)=(x^2-6x+12)/(x-4) is f(x) = 6

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range of f(x)=(x^2-6x+12)/(x-4)

range

f (x) = \frac{x ^{2} - 6 x + 12}{x - 4}

f (x) \leq - 2 or f (x) \geq 6

Solution steps

Rewrite as

\frac{x ^{2} - 6 x + 12}{x - 4} = y

Multiply both sides by

x - 4

x^{2} - 6 x + 12 = y (x - 4)

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

Discriminant

x^{2} - 6 x + 12 = y (x - 4) : y^{2} - 4 y - 12

y^{2} - 4 y - 12 \geq 0 : y \leq - 2 or y \geq 6

Check if the range interval endpoints are included

y = - 2 :

Included

y = 6 :

Included

Therefore the range is

f (x) \leq - 2 or f (x) \geq 6

6 x + y = - 1

7 - 2 x + 2

f (x) = sin (2 (x + π))

y = 6 - x - 4 x^{2} - x^{3}

f (x) = \frac{x - 7}{5 x ^{2}}

What is the range of f(x)=(x^2-6x+12)/(x-4) ?
The range of f(x)=(x^2-6x+12)/(x-4) is f(x)<=-2\lor f(x)>= 6