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

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

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

range

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

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

Interval Notation

(- \infty, - 2] \cup [6, \infty)

Solution steps

Rewrite as

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

Multiply both sides by

x - 2

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

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

Discriminant

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

- x + 2

f (x) = (x - 1) (x - 7)

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

\frac{x + 8}{x ^{2} - 5 x - 24}

\frac{1}{x} - 3

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