Let f be an invertible real function. Write

$(f^{- 1} o f) (1) + (f^{- 1} o f) (2) + . . . + (f^{- 1} o f) (100) .$

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Solution

Given that f is an invertible real function.
$f^{- 1} o f = I, where I is an identity function. So, (f^{- 1} o f) (1) + (f^{- 1} o f) (2) + . . . + (f^{- 1} o f) (100) = I (1) + I (2) + . . . + I (100) = 1 + 2 + . . . + 100 (As I (x) = x, \forall x \in R) = \frac{100 (100 + 1)}{2} [Sum of first n natural numbers= \frac{n (n + 1)}{2}] = 5050$

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