
7. Let $X$ be the set of all positive integers greater than or equal to 8 and let $f: X \rightarrow X$ be a function such that $f(x+y)=f(x y)$ for all $x \geq 4, y \geq 4$. If $f(8)=9$, determine $f(9)$.


Solution: We observe that

$$
\begin{aligned}
f(9)=f(4+5)=f(4 \cdot 5)= & f(20)=f(16+4)=f(16 \cdot 4)=f(64) \\
& =f(8 \cdot 8)=f(8+8)=f(16)=f(4 \cdot 4)=f(4+4)=f(8)
\end{aligned}
$$

Hence if $f(8)=9$, then $f(9)=9$. (This is one string. There may be other different ways of approaching $f(8)$ from $f(9)$. The important thing to be observed is the fact that the rule $f(x+y)=f(x y)$ applies only when $x$ and $y$ are at least 4. One may get strings using numbers $x$ and $y$ which are smaller than 4, but that is not valid. For example

$$
f(9)=f(3 \cdot 3)=f(3+3)=f(6)=f(4+2)=f(4 \cdot 2)=f(8)
$$

is not a valid string.)

