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Question
Let f be an invertible real function. Write ( f-1 of ) (1) + ( f-1 of ) (2) +..... +( f-1 of ) (100 )
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Solution
Given that f is an invertible real function.
\[f^{- 1} o f = I, \text{where I is an identity function}.\]
\[So,\]
\[\left( f^{- 1} o f \right)\left( 1 \right) + \left( f^{- 1} o f \right)\left( 2 \right) + . . . + \left( f^{- 1} o f \right)\left( 100 \right)\]
\[ = I\left( 1 \right) + I\left( 2 \right) + . . . + I\left( 100 \right)\]
\[ = 1 + 2 + . . . + 100 \left( AsI\left( x \right) = x, \forall x \in R \right)\]
\[ = \frac{100\left( 100 + 1 \right)}{2}[\text{ Sum of first n natural numbers}=\frac{n\left( n + 1 \right)}{2}]\]
\[ = 5050\]
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