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# Alg2 Help

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Find the inverse of the function $$f(x)=\frac{2^x}{1+2^x}$$

Mar 8, 2018

#1
+97543
+1

This is not correct       Can anyone spot what mistake I have made??

$$f(x)=\frac{2^x}{1+2^x}\\ let \\ y=\frac{2^x}{1+2^x}\\ \text{An initial inspection tells me that y>0}\\ \text{make x the subject}\\ y=\frac{2^x}{1+2^x}\\ \frac{1}{y}=\frac{1+2^x}{2^x}\\ \frac{1}{y}=\frac{1}{2^x}+\frac{2^x}{2^x}\\ \frac{1}{y}=\frac{1}{2^x}+1\\ \frac{1}{y}-1=2^{-x}\\ \frac{1-y}{y}=2^{-x}\\ log_2({\frac{1-y}{y}})=log_2(2^{-x})\\ log_2({\frac{1-y}{y}})=-x*log_2(2)\\ log_2({\frac{1-y}{y}})=-x\\ x=-log_2({\frac{1-y}{y}})\\ x=log_2(({\frac{1-y}{y}})^{-1})\\ x=-log_2\left({\frac{y}{1-y}}\right)\\ \therefore f^{-1}(x)=-log_2\left({\frac{x}{1-x}}\right)\\$$

I have graphed this here

https://www.desmos.com/calculator/qt6xarwxbp

AND IT IS NOT CORRECT

Can anyone spot what mistake I have made??

Mar 8, 2018
edited by Melody  Mar 8, 2018
#5
+97543
+3

Thanks Heureka,

I found my own error too :)

It was just a careless sign error in my last line.

$$x=log_2(({\frac{1-y}{y}})^{-1})\\ x=-log_2\left({\frac{y}{1-y}}\right) \qquad \text{This should have been } \color{red}{x=+log_2\left({\frac{y}{1-y}}\right) }\\ \therefore f^{-1}(x)=\color{red}{+} log_2\left({\frac{x}{1-x}}\right)\\ \text{Using the change of base rule this is exactly the same as Heureka's answer.}$$

Melody  Mar 8, 2018
#2
+68
+1

I tried it by myself an failed...

Then I used the internet and found the answer with steps.

I checked it out in Desmos and it worked. https://www.desmos.com/calculator/qt6xarwxbp

Thank yooooou internet. xD

Mar 8, 2018
#4
+97543
+1

BUT

Your desmos graph [  https://www.desmos.com/calculator/qt6xarwxbp ] is incorrect.

You have graphed it in base 2 NOT in base e.

As you can see, your two graphs are not reflections of one another over the line y=x

That is how I know it is the wrong answer (the answer that you have graphed is wrong I mean)

Melody  Mar 8, 2018
#3
+21242
+2

Find the inverse of the function

$$\displaystyle f(x)=\frac{2^x}{1+2^x}$$

f(x)=\frac{2^x}{1+2^x}

$$\begin{array}{|rcll|} \hline y &=& \dfrac{2^x}{1+2^x} \quad & | \quad \text{substitute u =2^x } \\\\ y &=& \dfrac{u}{1+u} \\\\ y(1+u) &=& u \\ y+yu &=& u \\ y &=& u-yu \\ y &=& u(1-y) \\\\ u&=& \dfrac{y}{1-y} \quad & | \quad \text{substitute u =2^x } \\\\ 2^x&=& \dfrac{y}{1-y} \quad & | \quad \text{ \ln{} both sides } \\\\ \ln(2^x) &=& \ln\left( \dfrac{y}{1-y} \right) \\\\ x\ln(2) &=& \ln\left( \dfrac{y}{1-y} \right) \\\\ x &=& \dfrac{ \ln\left( \dfrac{y}{1-y} \right) } {\ln(2) } \quad & | \quad x \Leftrightarrow y \\\\ \mathbf{f^{-1}} &\mathbf{=}&\mathbf{ \dfrac{ \ln\left( \dfrac{x}{1-x} \right) } {\ln(2) } } \\ \hline \end{array}$$

Mar 8, 2018
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