[tex]\sqrt x[/tex] is defined for [tex]x\ge0[/tex]. This means [tex]\sqrt{x+3}[/tex] exists only for [tex]x+3\ge0[/tex], or [tex]x\ge-3[/tex]; and [tex]\sqrt{5-x}[/tex] exists for [tex]5-x\ge0[/tex], or [tex]x\le5[/tex]. We can't divide by 0, so we have to exclude [tex]x=0[/tex] from the domain.
So the domain of [tex]f(x)[/tex] is [tex][-3,5]\setminus\{0\}[/tex], or [tex][-3,0)\cup(0,5][/tex].
