The function #f(x)# is defined for every positive integer #x# and satisfies the equation #f(x+y) = f(x)f(y) - f(xy) + 1# for all positive integers #x# and #y#. If #f(1) =2#, find #f(x)#?

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Answer 1 · 2017-06-01T14:39:48

#f(x)=x+1#

Making #y=1# we have

#f(x+1)=2f(x)-f(x)+1=f(x)+1#

or

#f(x+1)-f(x) = 1# then

#f(x) = x + C_0# but

#f(1)=1+C_0=2 rArr C_0=1# and finally

#f(x) = x + 1#