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Question

If a function f:ZZ is defined as mf(x)+f(my)=f(f(x+y)), where m is a non-zero constant, then which of the following is/are correct?

A
f(x)=mx+n, nZ
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B
f(x)=nx+m, nZ
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C
f(x)=0
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D
f(1),f(4),f(7),f(10) are in A.P.
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Solution

The correct options are
A f(x)=mx+n, nZ
C f(x)=0
D f(1),f(4),f(7),f(10) are in A.P.
mf(x)+f(my)=f(f(x+y)) ...(1)
At y=0mf(x)+f(0)=f(f(x)) ...(2)
Replace x by x+1 in eqn(2)
mf(x+1)+f(0)=f(f(x+1)) ...(3)
Now, put y=1 in eqn(1), we get
mf(x)+f(m)=f(f(x+1)) ...(4)
From eqn(3) and (4), we have
mf(x+1)+f(0)=mf(x)+f(m)
f(x+1)f(x)=f(m)f(0)m=Constant

Since, the difference of two consecutive terms f(x+1) and f(x) is constant.
Therefore, f(1),f(4),f(7),f(10) are in A.P.

Let f(x)=kx+n ; nZ
From eqn(1),
m(kx+n)+kmy+n=f[k(x+y)+n]km(x+y)+mn+n=k2(x+y)+kn+n

Comparing the coefficients of (x+y) and constant term, we get,
km=k2 and mn+n=kn+n
k=0,m
When, k=0mn+n=nn=0
When, k=mmn+n=mn+nn=n
n can be any value.

When k=0, n=0f(x)=0,
When k=m, nZf(x)=mx+n

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