This question was uploaded on 05/08/21 on social media accounts.
In this puzzle, a system of function is given and you have to find f(x). Sum of f(x) and f(1/(1-x)) is equal to x.
Give this question a try then check your answer with a solution.
(Shortcut Image Solution with Calculations is gives below in this post)
\[\color{blue} {\Rightarrow f\left(x\right)+f\left(\frac{1}{1-x}\right)=x----(1)}\]
Step 1:
Replace all x in equation 1 with (1/1-x)
\[\color{red} {\Rightarrow f\left(\frac{1}{1-x}\right)+f\left(\frac{x-1}{x}\right)=\frac{1}{1-x}----(2)}\]
Step 2:
Replace all x in equation 1 with (x-1/x)
\[\color{fuchsia} {\Rightarrow f\left(\frac{x-1}{x}\right)+f(x)=\frac{x-1}{x}----(3)}\]
Step 3:
Add equations (1) and (3)
\[\Rightarrow\color{blue} {f\left(x\right)+f\left(\frac{1}{1-x}\right)}+\color{fuchsia} {f\left(\frac{x-1}{x}\right)+f(x)}=\color{blue}{x}+\color{fuchsia}{\frac{x-1}{x}}\]
\[\Rightarrow f\left(x\right)+\color{red}{\frac{1}{1-x}}+f(x)=x+\frac{x-1}{x}---from (2)\]
\[\Rightarrow2f(x)=x+\frac{x-1}{x}-\frac{1}{1-x}=\frac{x^3-x+1}{x(x-1)}\]
\[\Rightarrow f(x)=\frac{x^3-x+1}{2x(x-1)}=\frac{x^3-x+1}{2x^2-2x}\]
Image Solution:
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