Given the vector space: $\ V:= [~~f:ℝ→ℝ∣ for ~~ a_0,…,a_4~∈~ℝ~and~ f(x)=∑^4_{i=0}a_ix^i ~~\forall ~x∈ℝ]$
and given the function:
$\ φ: V \rightarrow V, ~defined~ as~~ φ(f)(x)=f″(x)+x⋅f′(x)−f(x+1) $
How do I apply the inverted function $\ φ^{-1}$({g}) to the given function g ∈ V, where g is defined as $\ g(x)=3x^ 4+2x^3−x+1? $
Let $f\in V$ be defined by the scalars $a_0,\ldots ,a_4$. Then
$$\phi (f)(x)=-a_0-a_1+a_2-a_3-a_4+(-2a_2+3a_3-4a_4)x+(a_2-3a_3+6a_4)x^2+(2a_3-4a_4)x^3+3a_4x^4$$
for every $x\in \mathbb{R}$ (this is direct computation: expand everything then gather it together).
Thus, we have $f\in \phi ^{-1}(\{ g\} )$ if and only if the following system of equations is satisfied:
$$\begin{cases} -a_0-a_1+a_2-a_3-a_4=1 \\ -2a_2+3a_3-4a_4=-1 \\ a_2-3a_3+6a_4=0 \\ 2a_3-4a_4=2 \\ 3a_4=3 \end{cases} $$
Now, I let you solve this by yourself.