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Real number m,n,p,q such that f(g(x))=g(f(x)) has a solution

Source: 1976 AHSME Problem 10

May 15, 2014
AMC

Problem Statement

If m, n, p,m,~n,~p, and qq are real numbers and f(x)=mx+nf(x)=mx+n and g(x)=px+qg(x)=px+q, then the equation f(g(x))=g(f(x))f(g(x))=g(f(x)) has a solution
<spanclass=′latex−bold′>(A)</span>for all choices of m, n, p, and q<span class='latex-bold'>(A) </span>\text{for all choices of }m,~n,~p, \text{ and } q\qquad
<spanclass=′latex−bold′>(B)</span>if and only if m=p and n=q<span class='latex-bold'>(B) </span>\text{if and only if }m=p\text{ and }n=q\qquad
<spanclass=′latex−bold′>(C)</span>if and only if mq−np=0<span class='latex-bold'>(C) </span>\text{if and only if }mq-np=0\qquad
<spanclass=′latex−bold′>(D)</span>if and only if n(1−p)−q(1−m)=0<span class='latex-bold'>(D) </span>\text{if and only if }n(1-p)-q(1-m)=0\qquad
<spanclass=′latex−bold′>(E)</span>if and only if (1−n)(1−p)−(1−q)(1−m)=0<span class='latex-bold'>(E) </span>\text{if and only if }(1-n)(1-p)-(1-q)(1-m)=0
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