# Math Help - Finding domain for this piecewise function w/o calc

1. ## Finding domain for this piecewise function w/o calc

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 ???

2. ## Re: Finding domain for this piecewise function w/o calc

Originally Posted by snickerdoodle27
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) ]
But I'm confused w/ the DOMAIN! The answer sheet says it's ( -infinity, -2 ) u (2, infinity) ???? how ???
Do you understand how to determine the domain of $y=\frac{1}{x-2}~?$

3. ## Re: Finding domain for this piecewise function w/o calc

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?

4. ## Re: Finding domain for this piecewise function w/o calc

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.

5. ## Re: Finding domain for this piecewise function w/o calc

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!!

6. ## Re: Finding domain for this piecewise function w/o calc

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