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  4. If f(x) = (x/x-1), then ((fofo........of )(x)/19 times) is equal to :

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Q. If $f\left(x\right) = \frac{x}{x-1}$, then $\frac{\left(fofo........of \right)\left(x\right)}{19 \,times}$ is equal to :

Relations and Functions - Part 2

Solution:

Given $f\left(x\right) = \frac{x}{x-1}$
$\therefore \left(f\,o \,f\right) \left(x\right) = f \,{f \left(x\right)} = f\left(\frac{x}{x-1}\right)$
$= \frac{\frac{x}{x-1}}{\frac{x}{x-1}-1} = \frac{\frac{x}{x-1}}{\frac{x-x+1}{x-1}} = \frac{\frac{x}{x-1}}{\frac{1}{x-1}} = x$.
$\Rightarrow \left(f \,o\,f \,o\,f \right)\left(x\right) f \left(f \,o\,f \right)\left(x\right) f \left(x\right) = \frac{x}{x-1}$
$\Rightarrow \underbrace{\left(f \,o\,f\,o\,f ..... o\,f \right)} \left(x\right) = f \left(f\, o\, f\right)\left(x\right) = f \left(x\right) =\frac{x}{x-1}$
$\quad$$\quad$$\quad$19 times