Question

Find the domains of definition of the following functions.
$f\left(x\right)=\sqrt{9-x^2}+log(x-1)-\sqrt{x}.$

Answer 1

$f\left(x\right)=\sqrt{9-x^2}+\log (x-1)-\sqrt{x}$

For $f\left(x\right)$ to be defined-

$\left(i\right)$ $9-x^2\ge 0⟹x^2\le 9$

$⟹-3\le x\le 3$

$\left(ii\right)$ For $\log (x-1)$ to be defined

$x-1>0⟹x>1$

$\left(iii\right)$ For $\sqrt{x}$ to be defined

$x\ge 0$

Hence domain of $f\left(x\right)$ is $x\in (1,3]$