Hi all,
The math homework is to "use the piecewise definition of the absolute value function to help you define f in pieces." I understand this part of the problem. My troubles are with the domain--I just don't know how to algebraically find the correct answer.
The absolute value function is:
y = -x [ abs(2-x) ] <- that's -x multiplied by abs(2-x)
--------------
(x-2)
I turned it into this piecewise function:
y = { x
{ -x
But I'm confused w/ the DOMAIN! The answer sheet says it's ( -infinity, -2 ) u (2, infinity) ???? how ???
Yes, I do. I actually figured out why 2 is a relevant number in this problem. So, then the final answer would be:
f(x) = x, x>2
-x, x<2
Or do the inequality signs switch?
one easy way to check is to use a "test value" on either side of the crucial point, so for example at x = 3, we have:
y = (-3)|2 - 3|/(3 - 2) = (-3)|-1|/1 = (-3)(1) = -3, which is -x.
at x = 1, we have:
y = (-1)|2 - 1|/(1 - 2) = (-1)|1|/(-1) = |1| = 1, which is x.
but it is FAR better to actually understand what is happening "piecewise".
IF x < 2, THEN 2 - x > 0, so |2 - x| = 2 - x. so for ALL such x:
(-x)|2 - x|/(x - 2) = (-x)(2 - x)/(x - 2) = (-x)(-(x - 2))/(x - 2) = (-x)(-1) = x.
OTHERWISE, if x > 2, then 2 - x < 0, in which case |2 - x| = -(2 - x) = x - 2.
thus (-x)|2 - x|/(x - 2) = (-x)(x - 2)/(x - 2) = -x, for all x > 2.
when one is dealing with absolute values, it is important to keep careful accounting of the signs of each and every quantity involved in an expression.
Got it. So you basically have to set the abs value part equal to zero, with the respective inequality sign, then solve. That's why the inequalities signs change - because you're dividing by a negative number. Thank you SO much!!
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