Let f be an invertible real function. Write

Question:

Let f be an invertible real function. Write

Solution:

Given that f  is an invertible real function.

$f^{-1} o f=I$, where $\mathrm{I}$ is an identity function.

So,

$\left(f^{-1} o f\right)(1)+\left(f^{-1} o f\right)(2)+\ldots+\left(f^{-1} o f\right)(100)$

$=I(1)+I(2)+\ldots+I(100)$

$=1+2+\ldots+100($ As $I(x)=x, \forall x \in R)$

$=\frac{100(100+1)}{2}\left[\right.$ Sum of first n natural numbers $\left.=\frac{n(n+1)}{2}\right]$

$=5050$

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