Q:Let f:X-Y and g:y-z be two invertible functions.
Then gof is also invertible with
                    left parenthesis g o f right parenthesis space to the power of negative 1 end exponent=f to the power of negative 1 end exponentog to the power of negative 1 end exponent.

Asked by Ravi consul | 15th Mar, 2015, 07:47: PM

Expert Answer:

T o space p r o v e space t h a t space left parenthesis g o f right parenthesis to the power of negative 1 end exponent equals f to the power of negative 1 end exponent o g to the power of negative 1 end exponent comma space i t space i s space w o u l d space b e space s u f f i c i e n t space t o space s h o w space t h a t space left parenthesis g o f right parenthesis o left parenthesis f to the power of negative 1 end exponent o g to the power of negative 1 end exponent right parenthesis equals I subscript Y space a n d space left parenthesis f to the power of negative 1 end exponent o g to the power of negative 1 end exponent right parenthesis o left parenthesis g o f right parenthesis equals I subscript Z I subscript X comma I subscript Y space a n d space I subscript Z space a r e space t h e space i n v e r s e space e l e m e n t s space o f space X comma space Y space a n d space Z space r e s p e c t i v e l y. N o w comma space left parenthesis g o f right parenthesis o left parenthesis f to the power of negative 1 end exponent o g to the power of negative 1 end exponent right parenthesis equals space left parenthesis g o left parenthesis f o f to the power of negative 1 end exponent right parenthesis o g to the power of negative 1 end exponent right parenthesis space left square bracket S i n c e space left parenthesis h o g right parenthesis o f equals h o left parenthesis g o f right parenthesis right square bracket equals g o I subscript X o g to the power of negative 1 end exponent equals g o g to the power of negative 1 end exponent equals I subscript Y S i m i l a r l y comma space w e space c a n space s h o w space t h a t space left parenthesis f to the power of negative 1 end exponent o g to the power of negative 1 end exponent right parenthesis o left parenthesis g o f right parenthesis equals I subscript Z

Answered by Prasenjit Paul | 16th Mar, 2015, 11:47: AM

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