Question

Suppose g(x)=ax+b, where a and b are real constants. Find all possible pairs (a,b) such that we have g(g(x))=9x+28.
Show work or explain.

Answer 1

$g(x)=ax+b\implies g(g(x))=g(ax+b)=a(ax+b)+b=a^2x+ab+b$

You have $a^2=9\implies a=\pm3$, and $ab+b=28$. If $a=3$, then $3b+b=4b=28\implies b=7$. If $a=-3$, then $-3b+b=-2b=28\implies b=-14$.

So the possible pairs are $(3,7)$ and $(-3,-14)$.