Find $b$ and $c$ for $f(x)=\log(-x^2+bx+c) $

Question

I have a function $f(x)=\log(-x^{2}+bx+c) $ and the domain of $f$ is $(1,3)$. I have to use this fact to find the values of $b$ and $c$. I thought about solving the inequality $1 < \log(-x^2+bx+c) < 3 $ but not sure about that approach.

Answer 1

The inequality you have given would make the range of $f(x)$ be $(1,3)$. For the domain, you want $-x^2+bx+c>0$, and you would want to find $b$ and $c$ such that only for $x \in (1,3)$ would $-x^2+bx+c>0$.