Q.Find the domain and range of the function $f(x)=\sqrt{(x-2)(5-x)}$.

$f(x)=\sqrt{(x-2)(5-x)}$.

Domain: need $(x-2)(5-x)\ge0$. This product is a downward parabola in $x$ with roots $2$ and $5$, so it is $\ge0$ exactly between the roots: $2\le x\le5$. Domain $=[2,5]$.

Range: on $[2,5]$, write $(x-2)(5-x)=-x^2+7x-10$, maximised at $x=\dfrac72$ (the midpoint of $2$ and $5$, since the parabola is symmetric): value $=-\left(\dfrac72\right)^2+7\left(\dfrac72\right)-10=-\dfrac{49}{4}+\dfrac{49}{2}-10=\dfrac{49}{4}-10=\dfrac{9}{4}$. …

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