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Question 31314: I have no idea if I am posting this problem in the right area but here goes. Let f(x)=x+1/x, and g(x)=2/x. Find the composition f*g(x)=f(g(x)) and simplify. I put a star where the little circle was. I don't even know how to find that little cirle! Thank you for your help.
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! Let f(x)=x+1/x, and g(x)=2/x. Find the composition f*g(x)=f(g(x)) and simplify. I put a star where the little circle was. I don't even know how to find that little cirle!
f*g(x)=f(g(x)).....MEANS PUT G(X) IN PLACE OF X IN F(X)..TO SIMPLIFY MATTERS FIRST
PUT G(X)=Y...SO WE HAVE TO FIND F(Y) WHICH IS EASY....
F(X)=X+1/X...SO...PUT Y IN PLACE OF X...
F(Y)=Y+1/Y=(Y^2+1)/Y....
NOW WE PUT Y=G(X)=2/X..SINCE Y IS AN INTERMEDIATARY WE INTODUCED ,LET US REMOVE IT BY SUBSTITUTING IT WITH Y=2/X...SO WE GET
f*g(x)=f(g(x))=(Y^2+1)/Y={(2/X)^2+1}/(2/X)={(4/X^2)+1}/(2/X)
={(4+X^2)/X^2}/(2/X)=(4+X^2)*X/(2X^2)=(4+X^2)/(2X)....IS THE ANSWER....
I put a star where the little circle was. I don't even know how to find that little cirle!
BY THE WAY ,THAT LITTLE CIRCLE YOU WERE MENTIONING ABOUT IS A GENERAL SYMBOL TO REPRESENT ANY OPERATION LIKE ADDITION OR SUBTRACTION OR MULTIPLICATION OR DIVISION ETC....SINCE THEY HAVE INDIVIDUAL SYMBOLS LIKE +,-..ETC,THE IRCLE IS USED AS A GENERAL SYMBOL TO INDICATE ANY OPERATION AS IS DEFINED.HERE IT IS DEFINED AS EQUAL TO F OF G OF X=F(G(X))...OK....
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