What is the range of (2x+5)/(3x+1) ?

The range of (2x+5)/(3x+1) is f(x) 2/3

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range of (2x+5)/(3x+1)

range

\frac{2 x + 5}{3 x + 1}

f (x) < \frac{2}{3} or f (x) > \frac{2}{3}

Interval Notation

(- \infty, \frac{2}{3}) \cup (\frac{2}{3}, \infty)

Solution steps

The function range is the combined domain of the inverse functions

Find the inverse function(s) of:

\frac{2 x + 5}{3 x + 1}

Inverse of

\frac{2 x + 5}{3 x + 1} : \frac{5 - x}{3 x - 2}

\frac{5 - x}{3 x - 2}

Find the domain of each inverse function

Domain of

\frac{5 - x}{3 x - 2} : x < \frac{2}{3} or x > \frac{2}{3}

Combine the intervals

f (x) < \frac{2}{3} or f (x) > \frac{2}{3}

What is the range of (2x+5)/(3x+1) ?
The range of (2x+5)/(3x+1) is f(x)< 2/3 \lor f(x)> 2/3