Do you know how I could compute the inverse function of the following exponential sentence?
$$y=\dfrac{e^x}{1+2e^x}$$
Tell me more
×
Mathematics Stack Exchange is a question and answer site for
people studying math at any level and professionals in related fields. It's 100% free, no registration required.
|
|
$$ \text{exponentiate} \longrightarrow \text{multiply by 2}\longrightarrow\text{add 1} \longrightarrow \text{reciprocal}\longrightarrow\text{multiply by }e^2 $$ What gets done last gets undone first. So the inverse is: $$ \text{Divide by }e^2\longrightarrow\text{reciprocal} \longrightarrow\text{subtract 1} \longrightarrow \text{divide by 2}\longrightarrow\text{take logarithm} $$ I.e. if $y = \dfrac{e^2}{1+2e^x}$ then $x = \log_e\left(\dfrac{\dfrac{1}{y/e^2} - 1}{2}\right)$. (Then simplify.) LATER EDIT: Since you've now said you wanted $e^x$ in the numerator, here's that: $$ y=\frac{e^x}{1+2e^x} = \frac{1}{e^{-x}+2} $$ $$ \text{multiply by }-1 \longrightarrow \text{exponentiate} \longrightarrow \text{add 2} \longrightarrow\text{reciprocal} $$ So the inverse is: $$ \text{reciprocal} \longrightarrow\text{subtract 2}\longrightarrow\text{take logarithm} \longrightarrow\text{multiply by }-1 $$ (Two of the steps are their own inverses.) So $$ x = -\log_e\left(\frac1y-2\right) $$ (Then simplify.) |
|||||||||
|
|
$$y=\dfrac{e^2}{1+2e^x}$$ $$1+2e^x=e^2/y$$ $$2e^x=e^2/y-1$$ $$e^x=\frac{e^2/y-1}{2}$$ $$x=\ln{\frac{e^2/y-1}{2}}$$ |
|||
|
|
