What is the range of ((x-1)(x+4))/((x+1)(x-6)) ?

The range of ((x-1)(x+4))/((x+1)(x-6)) is -infinity <f(x)<infinity

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range of ((x-1)(x+4))/((x+1)(x-6))

range

\frac{( x - 1 ) ( x + 4 )}{( x + 1 ) ( x - 6 )}

- \infty < f (x) < \infty

Interval Notation

(- \infty, \infty)

Solution steps

Rewrite as

\frac{( x - 1 ) ( x + 4 )}{( x + 1 ) ( x - 6 )} = y

Multiply both sides by

(x + 1) (x - 6)

(x - 1) (x + 4) = y (x + 1) (x - 6)

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

Discriminant

(x - 1) (x + 4) = y (x + 1) (x - 6) : 49 y^{2} - 10 y + 25

49 y^{2} - 10 y + 25 \geq 0 :

True for all

y \in R

Therefore the range is

- \infty < f (x) < \infty

What is the range of ((x-1)(x+4))/((x+1)(x-6)) ?
The range of ((x-1)(x+4))/((x+1)(x-6)) is -infinity <f(x)<infinity