1976 AMC 12 Question 10
AMC12 · 1976 AMC 12
If $m,$ $n,$ $p$ and $q$ are real numbers and $f(x)=mx+n$ and $g(x)=px+q,$ then the equation $f(g(x))=g(f(x))$ has a solution
for all choices of $m,$ $n,$ $p$ and $q$
if and only if $m=p$ and $n=q$
if and only if $mq-np=0$
if and only if $n(1-p)-q(1-m)=0$
if and only if $(1-n)(1-p)$ ${}-(1-q)(1-m)=0$