Range of graph for modulus functions

Question

If a function is given as $|3x - x^2|$ with range $x\le 3$, how will I find the range for the graphs for $3x-x^2$ and $-(3x-x^2)$ ?

Answer 1

Differentiate 3x-x^2 which is a downward facing parabola. You get 2x=3 or x=3/2. At x=3/2 the function is greatest so it's range is y<=9/4

Answer 2

$y=3x-x^2$ is a parabola, concave, with a maximum in $x=\dfrac32$. This function cuts the $x$-axis in $x=0$ and $x=3$, so when we get absolute $|3x-x^2|$ of it, two downward branches of it turn upward. The minimum of $|3x-x^2|$ occur in $x=0$ that is $0$ and left branch of it goes $+\infty$, thus the range of it is $[0,\infty)$. For graph of this function see WolframAlpha

Answer 3

$$3x-x^2=x(3-x)$$

$$|x(3-x)|= \begin{cases}+x(3-x) &\mbox{if }x(3-x)\ge0\iff x(x-3)\le0\iff0\le x\le3 \\-x(3-x)& \mbox{otherwise } \end{cases}$$