What is the range of y=(2x+3)/(4x+1) ?

The range of y=(2x+3)/(4x+1) is f(x) 1/2

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range of y=(2x+3)/(4x+1)

range

y = \frac{2 x + 3}{4 x + 1}

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{2 x + 3}{4 x + 1}

Inverse of

\frac{2 x + 3}{4 x + 1} : \frac{3 - x}{2 ( - 1 + 2 x )}

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

Find the domain of each inverse function

Domain of

\frac{3 - x}{2 ( - 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 y=(2x+3)/(4x+1) ?
The range of y=(2x+3)/(4x+1) is f(x)< 1/2 \lor f(x)> 1/2