A function $f$ satisfies $f(x)+f(x+3)=2 x+5$ for all $x$. If $f(8)+f(2)=12$, determine the value of $f(5)$.
Solution 1

Since $f(x)+f(x+3)=2 x+5$ for all $x$, then using $x=2$ we obtain $f(2)+f(5)=2(2)+5$ and so $f(2)+f(5)=9$.

Also, using $x=5$ we obtain $f(5)+f(8)=2(5)+5$, and so $f(5)+f(8)=15$.

Adding these two equations, we obtain $f(2)+f(8)+2 f(5)=24$.

Since $f(8)+f(2)=12$, then $2 f(5)=24-(f(2)+f(8))=24-12=12$ and so $f(5)=6$.

## Solution 2

The function $f(x)=x+1$ satisfies the required properties, since

- $f(x+3)=(x+3)+1=x+4$ and so $f(x)+f(x+3)=(x+1)+(x+4)=2 x+5$, and
- $f(2)=2+1=3$ and $f(8)=8+1=9$ gives $f(8)+f(2)=12$.

So $f(x)=x+1$ is a function that works and the question implies that the value of $f(5)$ is the same no matter which function $f$ that satisfies the given properties is chosen.

Therefore, $f(5)=5+1=6$.

(Note that there may be other functions that satisfy these relationships.)