If $f\colon\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function satisfying $f(x)\cdot f(f(x)) = 1$ for all $x \in \mathbb{R},$ and if $f(10) = 9$ then find the value of $f(5).$
Attempt: Put $x=10,$ we have $f(10)\cdot f(f(10)) = 1.$
So, $9\cdot f(9) = 1,$ then $f(9) = \frac{1}{9}.$
Could someone help me how calculate $f(5)$ ? Thanks.
It follows from the assumption that for every $y$ in the image of $f$ we have$$f(y)=\frac{1}{y}.$$Now, as you already found out yourself, both $9$ and $1/9$ are in the image. As $f$ is assumed to be continuous, it follows that $5$ is also in the image, and so,$$f(5)=\frac{1}{5}.$$
You have calculated $f(9)=1/9$. So consider the interval $[9,10]$. Since $f$ is continuous, by intermediate value theorem, you can find $z$, which lies in $[9,10]$ such that $f(z)=5$.
Now $f(z)\cdot f(f(z))=5 \cdot f(5)=1$. Then $f(5)=1/5.$
