Piecewise Notation and Absolute Value

Date: 9/9/96 at 13:25:22
From: Anonymous
Subject: Piecewise Notation and Absolute Value

Using piecewise notation, write x in terms of y for the following 
equation:  "y = 2x + abs(2-x)".  I have graphed the equation and I 
think part of the answer is x = y-2 for y in the range of negative 
infinity to 4. Is there another way to do this rather than graphing? 
Please describe the method.

Date: 02/04/97 at 22:27:01
From: Doctor Sam
Subject: Re: Piecewise Notation and Absolute Value

If you already graphed y = 2x + abs(2-x) then you know that it looks 
like parts of two lines.  Those parts are the "pieces" in "piecewise 
notation." The reason that there are two pieces is that the absolute 
value function is also in pieces:

                x when x is positive or zero
   abs (x) =
               -x when x is negative

To change your absolute value equation into piecewise form ask 
yourself, "When is 2-x positive or zero?  When is it negative?"

Since 2-x = 0 when x = 2, this point marks the x-value of the part of 
the function when the graph changes from one piece to the next.   
When x < 2 the expression inside abs(...) is a positive number so  
abs(2 - x) = 2 - x.

On the other hand, when x > 2 the expression inside abs( ) is a 
negative number and so its absolute value is the opposite.
That is, abs(2 - x) = - (2 - x) = x - 2 for x-values in this
range.

To finish the problem, figure out what y is in each part of
the graph:

            /  2x + (2 - x)    when x < 2   \
       y =  |                                |
            \  2x + (x - 2)    when x >=2   /

Finally, simplify these two expressions to get that y = x + 2
when x < 2 and y = 3x-2 when x >=2.

I hope that helps!

Write back if you need more help.

--Doctors Sam and Sydney, The Math Forum