Find the image of an element without clear function rule. How can I find $f(4)$ from the following equation:

$$f(x) + f(2x+1) = 6x+10$$
Let $g(x)=f(x)-2x-4$ to get

$$g(x)+g(2x+1)=0$$

Substitute $x\mapsto2^x-1$ and let $h(x)=g(2^x-1)$ to get

$$h(x)+h(x+1)=0$$

We can produce infinitely many solutions for $h$. An obvious form can be given by trigonometric functions, and more generally Fourier series. As a simple example:

$$h(x)=A\sin(\pi x)$$

$$g(x)=A\sin(\pi\log_2(x+1))$$

$$f(x)=2x+4+A\sin(\pi\log_2(x+1))$$

$$f(4)=12+A\sin(\pi\log_2(5))$$

which depends on the arbitrary value $A$. Thus there is no unique solution, given the conditions.
$f(x)=2x+4$ satisfies the equation so $f(4)=12$