10.4. Find all functions $f(x)$ defined on the entire real line that satisfy the equation $f(y-f(x))=1-x-y$ for any $x$ and $y$.
Answer. $f(x)=\frac{1}{2}-x$.

Solution. 1) First, let $y=f(x)$ in the condition, then $f(x)=1-x-f(0)$.

2) Substitute the obtained expression for $f(x)$ into the condition, then: $f(y-f(x))=1-y+f(x)-f(0)=1-y+1-x-f(0)-f(0)=1-x-y, \quad$ from which $\quad f(0)=\frac{1}{2}$.

Therefore, $f(x)=\frac{1}{2}-x$ is the only candidate for the solution.

3) Verification by substitution: $\quad f(y-f(x))=\frac{1}{2}-y+f(x)=\frac{1}{2}-y+\frac{1}{2}-x=1-x-y \quad-$ the condition of the problem is satisfied, therefore, $f(x)=\frac{1}{2}-x$ is the only solution to the problem.

Grading criteria. Found only $f(0)=\frac{1}{2}: 2$ points. Obtained the relation $f(x)=1-x-f(0): 3$ points. Lack of verification: minus 1 point.