Express the following equation as a piecewise: f(x)= |x^2 -4x+3| Ok so I know f(x)= |(x-1)(x-3)| therefore the piece wise "splits" at x=1 and x=3. However, I am confused as to which intervals I would make f(x) positive or negative. Is there a rule for figuring this out?
2026-03-25 17:39:43.1774460383
Turning an absolute value of a polynomial into a piece wise
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Generally $|x|=\left \{\begin {array}{ll} x,~x\geq 0\\ -x,~x\leq 0\\ \end{array} \right..~~$ So $$|x-1|=\left \{\begin {array}{ll} x-1,~x\geq 1\\ -(x-1),~x\leq 1\\ \end{array} \right.,~~|x-3|=\left \{\begin {array}{ll} x-3,~x\geq 3\\ -(x-3),~x\leq 3\\ \end{array} \right.$$ and
$$f(x)=|x-1|~|x-3|=\left \{\begin {array}{ll} (x-1)(x-3),~x\geq 3\\ -(x-1)(x-3),~1\leq x\leq 3\\ (x-1)(x-3),~x\leq 1\\ \end{array} \right..$$