Preselection tests for the full-time training

Let $x_0$ be a solution of the equation $f(x)=x$. We have $$x_0=f(x_0)=f(f(x_0))=4x_0+1.$$ Therefore, $x_0=-\frac{1}{3}$. This proves the uniqueness of the solution.

On the other hand, $$f\left(f\left(-\frac{1}{3}\right)\right)=4\left(-\frac{1}{3}\right)+1=-\frac{1}{3}$$ We deduce that $$f\left(-\frac{1}{3}\right)=f\left(f\left(f\left(-\frac{1}{3}\right)\right)\right)=4f\left(-\frac{1}{3}\right)+1$$ and therefore, $$f\left(-\frac{1}{3}\right)=-\frac{1}{3}$$ This proves the existence of the solution.