Expert: Sherry Wallin - 2/12/2010

Question
simplify  x-|x-x^2|,  -1<=x<=1

let y=x-|x-x^2|
`
now |x-x^2|=x-x^2 >=0
i.e.     if   x(1-x)>=0
i.e.     if   x(x-1)<=0
i.e.     if   0<=x<=1

y=x-(x-x^2)=x^2,   if 0<=x<=1

Again  |x-x^2|=-(x-x^2) if x-x^2<0
i.e.     if   x(1-x)<0
i.e.     if   x(x-1)>0
i.e      if   x>1   or x<0

taking values from the given
i.e.     if   -1<=x<=1

Heance
y=x-{-(x-x^2)}=x+x-x^2
             =2x-x^2,    if  -1<=x<0

y=x-|x-x^2|=   {x^2     if 0<=x<=1
              {2x-x^2,  if   -1<=x<0

sherry please explain it

Answer
now |x-x^2|=x-x^2 >=0
i.e.     if   x(1-x)>=0  (1)
i.e.     if   x(x-1)<=0  (2)
i.e.     if   0<=x<=1    (3)

What this part is saying is that if the absolute value of x-x^2 is greater than or equal to zero, i.e., positive then (1) follows because it is just factoring x-x^2= x(1-x). If that value is greater than or equal to zero then changing the sign from 1-x to x-1 must be less than or equal to zero. x(x-1)<=0 implies either(or both) that x = 0 r x-1 = 0 -> x = 1, hence 0<=x<=1

y=x-(x-x^2)=x^2,   if 0<=x<=1

note anything squared is always positive and if x is between 0 and 1 inclusive then x is positive so under those conditions x-|x-x^2| is x-(x-x^2) does equal x^2

Again  |x-x^2|= -(x-x^2) if x-x^2<0 (4)
i.e.     if   x(1-x)<0              (5)
i.e.     if   x(x-1)>0              (6)
i.e      if   x>1   or x<0          (7)

(4) is the other case if |x-x^2| <=0
which means (5) that x(1-x) < 0
which means (6) x(x-1) > 0 because they've changed signs on 1-x to x-1
so(7) either x > 0 or x -1 > 0 -> x > 1 but by virtue of the hypothesis of the problem, x is at most 1 so it can't be greater than 1. Notice in order for x(1-x) >< 0 to be true either x has to be negative or 1-x has to be negative and not both. When 1-x < 0 then this implies that x is greater than one which is not in the domain of the original hypothesis so it must be that x < 0.

Heance
y=x-{-(x-x^2)}=x+x-x^2                      (8)
            =2x-x^2,    if  -1<=x<0

y=x-|x-x^2|=   {x^2     if 0<=x<=1          (9)
             {2x-x^2,  if   -1<=x<0

(8) says that since in case (4) -(x-x^2) the x -[-(x-x^2)]= x+(x-x^2) = 2x-x^2
(9) just sums up the conclusion stating that when x is nonnegative then the value is x^2 and when x is negative up to and including -1 then the expression is equal to 2x-x^2

Math Prof