What is the range of (3x+6)/(-6x+2) ?

The range of (3x+6)/(-6x+2) is f(x) -1/2

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range of (3x+6)/(-6x+2)

range

\frac{3 x + 6}{- 6 x + 2}

f (x) < - \frac{1}{2} or f (x) > - \frac{1}{2}

Interval Notation

(- \infty, - \frac{1}{2}) \cup (- \frac{1}{2}, \infty)

Solution steps

The function range is the combined domain of the inverse functions

Find the inverse function(s) of:

\frac{3 x + 6}{- 6 x + 2}

Inverse of

\frac{3 x + 6}{- 6 x + 2} : - \frac{6 - 2 x}{3 ( 1 + 2 x )}

- \frac{6 - 2 x}{3 ( 1 + 2 x )}

Find the domain of each inverse function

Domain of

- \frac{6 - 2 x}{3 ( 1 + 2 x )} : x < - \frac{1}{2} or x > - \frac{1}{2}

Combine the intervals

f (x) < - \frac{1}{2} or f (x) > - \frac{1}{2}

What is the range of (3x+6)/(-6x+2) ?
The range of (3x+6)/(-6x+2) is f(x)<-1/2 \lor f(x)>-1/2