Question

Let $f\left(x\right)$ be a continuous function defined for $1\le x\le 3$. If $f\left(x\right)$ takes rational values for all $x$ and $f\left(2\right)=9$, then $f\left(1.5\right)=$

Answer 1

Given: $f\left(x\right)$ is a continous function for $1\le x\le 3$

$f\left(x\right)$ takes rational value for all x $f\left(2\right)=9$

To find: $f\left(1.5\right)=?$

Sol: $f\left(x\right)$ is continous for $x\in [1,3]$

We can see that there will be some rational and remaining irrational points $[0,1]$

Between two rational numbers there exist infinitely many rational numbers

$f\left(x\right)$ can take only rational values

This is possible only when $f\left(x\right)$ is constant for $x\in [1,3]$

$f\left(2\right)=9$

$⟹f\left(x\right)=9$ for all $x\in [1,3]$

$⟹f\left(1.5\right)=9$

Hence correct answer is $9$