Calculation with function For as long as I go around it I do not get to anything
If  $\phi \left ( f(x)-1\right )=2x+5$
and
$\phi (x)=2f(x+1)+1$
find $f(4)$

(Too long for a comment.)  Let $\,\varphi(x)=\phi(x-1)\,$, then the equations can be written as:
$$
\begin{align}
\varphi(f(x)) &= 2x+5 \tag{1} \\
\varphi(x) &= 2 f(x) + 1 \tag{2}
\end{align}
$$
Substituting $x \mapsto f(x)$ in $(2)$ and equating to $(1)$ gives:
$$\varphi(f(x)) = 2 f(f(x)) + 1=2x+5 \quad\implies\quad f(f(x)) = x+2 \tag{3}$$
Substituting $x \mapsto f(x)$ in $(1)$ and using $(2),(3)$ gives:
$$\varphi(x+2) = \varphi(f(f(x)))=2 f(x)+5 = (\varphi(x)-1)+5=\varphi(x)+4 \tag{4}$$
Per $(2)$ the latter is equivalent to:
$$2f(x+2)+1 = 2f(x)+1+4 \quad\iff\quad f(x+2)=f(x)+2 \tag{5}$$
P.S.  By inspection $f(x)=x+1$ and $\varphi(x)=2x+3 \iff \phi(x)=2x+5$ satisfy the functional equations, so if the problem has a unique answer then that must be $f(4)=5$.
