What is the range of f(x)=(x^2+x+1)/(x^2-7x+12) ?

The range of f(x)=(x^2+x+1)/(x^2-7x+12) is f(x) = 2sqrt(273)-33

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range of f(x)=(x^2+x+1)/(x^2-7x+12)

range

f (x) = \frac{x ^{2} + x + 1}{x ^{2} - 7 x + 12}

f (x) \leq - 2273 - 33 or f (x) \geq 2273 - 33

Interval Notation

(- \infty, - 2273 - 33] \cup [2273 - 33, \infty)

Solution steps

Rewrite as

\frac{x ^{2} + x + 1}{x ^{2} - 7 x + 12} = y

Multiply both sides by

x^{2} - 7 x + 12

x^{2} + x + 1 = y (x^{2} - 7 x + 12)

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

Discriminant

x^{2} + x + 1 = y (x^{2} - 7 x + 12) : y^{2} + 66 y - 3

y^{2} + 66 y - 3 \geq 0 : y \leq - 2273 - 33 or y \geq 2273 - 33

Check if the range interval endpoints are included

y = - 2273 - 33 :

Included

y = 2273 - 33 :

Included

Therefore the range is

f (x) \leq - 2273 - 33 or f (x) \geq 2273 - 33

(\frac{1}{2}, 4), (3, \frac{1}{4})

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

\frac{x - 2}{x - 1}

f (x) = 2 x^{3} + 1

f (x) = (x - 2) (x - 5)^{3} + 4

What is the range of f(x)=(x^2+x+1)/(x^2-7x+12) ?
The range of f(x)=(x^2+x+1)/(x^2-7x+12) is f(x)<=-2sqrt(273)-33\lor f(x)>= 2sqrt(273)-33