1976 AMC 12 Problem 10

Attempt Problem 10 of the 1976 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1976 AMC 12 solutions, or check the answer key.

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10.

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

Expand both compositions before trying to solve for

x

The coefficients of

x

are automatically equal, leaving a condition on the constant terms

We have

\begin{aligned} f(g(x))&=mpx+mq+n,\\ g(f(x))&=pmx+pn+q. \end{aligned}

The

x

-terms cancel for every

x,

so a solution exists exactly when

mq+n=pn+q.

Rearranging gives

n(1-p)-q(1-m)=0.

Therefore, the correct answer is D.

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