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Let `A = R - {(3)/(5)} " and " B =R -{(7)/(5)}` Let `f: A to B : f (x) =(7x +4)/(5x-3) " and " g: B to A : g (y)= (3y+4)/(5y-7)` Show that `(g o f) I_(A) " and " ( f o g) = I_(B)` |
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Answer» Let `x in A. `Then (g o f) (x) = g [f (x)] `=g ((7x+4)/(5x-3))` `=g(y) " where " y =(7x+4)/(5x-3)` `=(3y+4)/(5y-7) =(3((7x+4)/(5x-3))+4)/(5((7x+4)/(5x-3))-7) " " "[using (i)]"` `=((21 x+12 +20x-12))/((5x-3)) xx ((5x-3))/((35 x+20 -35 x+21))` `=(41x)/(41) =x= I_(A) (x)` `:. (g o f) =I_(A)` Again let `y in B .` Then ( f o g) (y) = f[g (y)] `=f((3y+4)/(5y-7))` `=f(z) " where " z= (3y+4)/(5y-7)` `=(7z+4)/(5z-3) =(7((3y+4)/(5y-7))+4)/(5((3y+4)/(5y-7))-3)` `=((21 y+28 +20y-28))/((5y-7)) xx ((5y-7))/((15y+20 -15y+21))` `=(41y)/(41) =y= I_(B) (y)` `:. (f o g) =I_(B)` hence (f o g ) `=I_(A) " and " (f o g) =I_(B)` |
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