Consider the following exercise:
Given that $g\left(x-3\right)=x^4$ solve the equation $$g\left(x\right)-81=g\left(x+3\right)+108x$$
This appears in a textbook in a section about Combinatorics and Binomial Coefficients. Prior to this exercise there is no obvious reference to this kind of problem.
In the solution for this problem, the original equation is promptly substituted by $\left(x+3\right)^4-81=x^4+108x$. My question is: how did it go from $g\left(x\right)$ to $\left(x+3\right)^4$?
Write $t=x-3$ then $t=x+3$, so $$g(t)=g(x-3)= x^4= (t+3)^4$$
Since $x\mapsto x-3$ is surjective you can write again it with $x$, so $g(x)=(x+3)^4$.