What is the range of 1/(5+e^{3x)} ?

The range of 1/(5+e^{3x)} is 0<f(x)< 1/5

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range of 1/(5+e^{3x)}

range

\frac{1}{5 + e ^{3 x}}

0 < f (x) < \frac{1}{5}

Interval Notation

(0, \frac{1}{5})

Solution steps

The function range is the combined domain of the inverse functions

Find the inverse function(s) of:

\frac{1}{5 + e ^{3 x}}

Inverse of

\frac{1}{5 + e ^{3 x}} : \frac{ln ( \frac{1}{x} - 5 )}{3}

\frac{ln ( \frac{1}{x} - 5 )}{3}

Find the domain of each inverse function

Domain of

\frac{ln ( \frac{1}{x} - 5 )}{3} : 0 < x < \frac{1}{5}

Combine the intervals

0 < f (x) < \frac{1}{5}

f (x) = 9 x + 13

f (x) = - \frac{1}{2} x - 2

f (x) = \frac{8 x ^{2} + 1}{4 x ^{2} + 2 x - 6}

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

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