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

The range of (x-3)/(x^2+x+3) is (-2sqrt(15)-7)/(11)<= f(x)<= (2sqrt(15)-7)/(11)

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

range

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

\frac{- 2 15 - 7}{11} \leq f (x) \leq \frac{2 15 - 7}{11}

Interval Notation

[\frac{- 2 15 - 7}{11}, \frac{2 15 - 7}{11}]

Solution steps

Rewrite as

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

Multiply both sides by

x^{2} + x + 3

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

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

Discriminant

x - 3 = y (x^{2} + x + 3) : - 11 y^{2} - 14 y + 1

- 11 y^{2} - 14 y + 1 \geq 0 : \frac{- 2 15 - 7}{11} \leq y \leq \frac{2 15 - 7}{11}

Check if the range interval endpoints are included

y = \frac{- 2 15 - 7}{11} :

Included

y = \frac{2 15 - 7}{11} :

Included

Therefore the range is

\frac{- 2 15 - 7}{11} \leq f (x) \leq \frac{2 15 - 7}{11}

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

x - 2 + \frac{x ^{2}}{x ^{2} - 9}

f (x) = \frac{( 3 x ^{2} - 2 x + 7 )}{2 x + 5}

f (x) = \frac{9}{x + 2}

f (x) = 16 + x^{2} - x + 1

What is the range of (x-3)/(x^2+x+3) ?
The range of (x-3)/(x^2+x+3) is (-2sqrt(15)-7)/(11)<= f(x)<= (2sqrt(15)-7)/(11)