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Q.
If $f ( x )=\frac{ x }{ x -1}$, then $\frac{(\text { fofo........of })( x )}{19 \text { times }}$ is equal to:
Relations and Functions - Part 2
Solution:
$\because f ( x )=\frac{ x }{ x -1}$
$ \therefore ($ fo $f )( x )= f \{ f ( x )\}= f \left(\frac{ x }{ x -1}\right)$
$=\frac{\frac{ x }{ x -1}}{\frac{ x }{ x -1}-1}=\frac{\frac{ x }{ x -1}}{\frac{ x - x +1}{ x -1}}=\frac{\frac{ x }{ x -1}}{\frac{1}{ x -1}}= x .$
$\Rightarrow ($ fof of $)( x )= f ( f$ of $)( x )$
$= f ( x )=\frac{ x }{ x -1}$
$\Rightarrow \underbrace{( f \text { of of } \ldots .0 f )}_{19 \text { times }}( x )= f ($ fo $f )( x )$
$= f ( x )=\frac{ x }{ x -1}$