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

The range of f(x)=(-x^2)/(x+1) is f(x) = 4

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

range

f (x) \leq 0 or f (x) \geq 4

Interval Notation

(- \infty, 0] \cup [4, \infty)

Solution steps

Rewrite as

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

Simplify

\frac{- x ^{2}}{x + 1} : - \frac{x ^{2}}{x + 1}

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

Multiply both sides by

x + 1

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

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

Discriminant

- x^{2} = y (x + 1) : y^{2} - 4 y

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

Check if the range interval endpoints are included

y = 0 :

Included

y = 4 :

Included

Therefore the range is

f (x) \leq 0 or f (x) \geq 4

What is the range of f(x)=(-x^2)/(x+1) ?
The range of f(x)=(-x^2)/(x+1) is f(x)<= 0\lor f(x)>= 4