I have one question just want to be sure that I am correct.
Suppose we have two function $f(x)$ and $h(x)$, such that $f(x)$ is linear (i.e., $f(x)=m x+b$) and $h(x)$ is quadratic ($h(x)=ax^2+bx+c$). We are asked to find a $g(x)$ so that $f(g(x))$ is equal to $h(x)$.
I think that we first should determine what is the highest degree function and and for answer choose this kind of function (general form), put it into the given function and calculate coefficient. For example, consider two situation. In the first one, we are given $g(x)=2x+1$ and $h(x)=4x^2+4x+7$, we should find such $f$ so that $f(g(x))=h(x)$. For the second situation, we have $f(x)=3x+5$ and $h(x)=3x^2+3x+2$; find $g$ such that $f(g(x))=h$.
I think for first because degree of $h$ is $2$, we should choose $f$ as $ax^2+bx+c$; and for second also, quadratic form.
Am I correct? Please if something is incorrect in my logic, please inform me.