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

The range of f(x)=(x^2-2x-3)/(x+2) is f(x) = 2sqrt(5)-6

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

range

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

f (x) \leq - 25 - 6 or f (x) \geq 25 - 6

Interval Notation

(- \infty, - 25 - 6] \cup [25 - 6, \infty)

Solution steps

Rewrite as

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

Multiply both sides by

x + 2

x^{2} - 2 x - 3 = 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 - 3 = y (x + 2) : y^{2} + 12 y + 16

y^{2} + 12 y + 16 \geq 0 : y \leq - 25 - 6 or y \geq 25 - 6

Check if the range interval endpoints are included

y = - 25 - 6 :

Included

y = 25 - 6 :

Included

Therefore the range is

f (x) \leq - 25 - 6 or f (x) \geq 25 - 6

What is the range of f(x)=(x^2-2x-3)/(x+2) ?
The range of f(x)=(x^2-2x-3)/(x+2) is f(x)<=-2sqrt(5)-6\lor f(x)>= 2sqrt(5)-6