What is the range of (x+3)/((x+6)(x-1)) ?

The range of (x+3)/((x+6)(x-1)) is -infinity <f(x)<infinity

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range of (x+3)/((x+6)(x-1))

range

\frac{x + 3}{( x + 6 ) ( x - 1 )}

- \infty < f (x) < \infty

Interval Notation

(- \infty, \infty)

Solution steps

Rewrite as

\frac{x + 3}{( x + 6 ) ( x - 1 )} = y

Multiply both sides by

(x + 6) (x - 1)

x + 3 = y (x + 6) (x - 1)

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

Discriminant

x + 3 = y (x + 6) (x - 1) : 49 y^{2} + 2 y + 1

49 y^{2} + 2 y + 1 \geq 0 :

True for all

y \in R

Therefore the range is

- \infty < f (x) < \infty

f (x) = \frac{( x - 1 )}{x + 3}

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

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

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

f (x) = x^{3} - 5 x

What is the range of (x+3)/((x+6)(x-1)) ?
The range of (x+3)/((x+6)(x-1)) is -infinity <f(x)<infinity