## Task 3 - 100923

Given are two real numbers $m \neq 0$ and $n$. Furthermore, let $f$ be the function defined by $f(x)=m x+n$ for all real numbers.

a) Determine all numbers $x_{0}$ for which $2 \cdot f\left(x_{0}\right)=f\left(x_{0}+2\right)$ holds (i.e., for which the function value at $x_{0}+2$ is twice as large as the one at $x_{0}$) when $m=1$ and $n=0$!

b) Determine all numbers $x_{0}$ for which $2 \cdot f\left(x_{0}\right)=f\left(x_{0}+2\right)$ holds for any given real numbers $m \neq 0$ and $n$!
a) By choosing $m$ and $n$, $f(x)=x$ for all real numbers $x$, i.e., the solutions of the equation $2 \cdot x_{0}=x_{0}+2$ are sought, which is satisfied exactly for $x_{0}=2$.

b) Here the equation $2 m \cdot x_{0}+2 n=m \cdot x_{0}+2 m+n$ is to be solved, which is equivalent to $m \cdot x_{0}=2 m-n$, thus $x_{0}=2-\frac{m}{n}$.