SOLUTION: Suppose the functions f, g, h, r and l are defined as follows: g (x) = sqrt((x − 2)^2) − x/2 Write down Dg and solve the equation g (x) = 0.

Question 1142652: Suppose the functions f, g, h, r and l are defined as follows:
g (x) = sqrt((x − 2)^2) − x/2
Write down Dg and solve the equation g (x) = 0.

Answer by ikleyn(52915) (Show Source): You can put this solution on YOUR website!
.

For your attention :

    THERE ARE NO functions f, h, r, and l in this post.


    There is ONLY function g(x), and for it, the solution is as follows.

Solution

The function g(x) is defined as


    g(x) =  - ,


and the equation  g(x) = o  is   


     -  = 0.      (1)


The domain is the set of all real numbers.


In this domain, the given equation is equivalent to


     = ,


which, in turn, is equivalent to


    | x - 2 | = .      (2)


Vertical lines denote the absolute value.


To solve the equation (2), we consider two cases.


Case 1.  x >= 2.   In this case

         | x - 2 | = x-2;  therefore, equation (2) takes the form
    
           x - 2   = .

     It is simplified and solved in this way

           2x - 4 = x

            x = 4.



Case 2.  x < 2.   In this case

         | x - 2 | = -(x-2) = -x + 2;  therefore, equation (2) takes the form
    
           -x + 2   = .

     It is simplified and solved in this way

           -2x + 4 = x

            4 = 3x

            x = 



ANSWER.  The given equation has 2 (two, TWO) solutions :  x= 4  and  x= .


         You can check it on your own by substituting these values of x into the original equation.

Solved, answered, explained, checked and completed.

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