Problem (Algebra):
If $f(x)=x^{2}-x+2, g(x)=a x+b$, and $f(g(x))=9 x^{2}-3 x+2$, determine all possible ordered pairs $(a, b)$ which satisfy this relationship.

Solution:
First, we compute an expression for the composition of the two given functions:

$$
\begin{aligned}
f(g(x)) & =f(a x+b) \\
& =(a x+b)^{2}-(a x+b)+2 \\
& =a^{2} x^{2}+2 a b x+b^{2}-a x-b+2 \\
& =a^{2} x^{2}+(2 a b-a) x+\left(b^{2}-b+2\right)
\end{aligned}
$$

But we already know that $f(g(x))=9 x^{2}-3 x+2$, so comparing coefficients, we see that

$$
\begin{aligned}
a^{2} & =9 \\
2 a b-a & =-3 \\
b^{2}-b+2 & =2
\end{aligned}
$$

From the first equation, $a=3$ or $a=-3$.

From the third equation, $b^{2}-b=b(b-1)=0$ so $b=0$ or $b=1$.

There are thus 4 possible pairs $(a, b)$ which could solve the problem. We will check which pairs work by looking at the second equation.

From the second equation, $a(2 b-1)=-3$, so if $a=3$ then $b=0$, and if $a=-3$ then $b=1$. Therefore, the possible ordered pairs $(a, b)$ are $(3,0)$ and $(-3,1)$.

Final answer: $(3,0),(-3,1)$