SOLUTION: Please may you help me solve:f(x)=|6-2x|+|X-1|-2X giving the necessary arguments before graphing.what is the range of the function?whenf(x)>0?

The critical points are 6-2x = 0, i.e. x = 3;  and x-1=0, i.e. x = 1.


If x < 1, then        6 - 2x > 0 and hence |6-2x} = (6-2x);

                      x - 1 < 0  and hence |x-1| = (1-x);

                      thus the entire function is f(x) = (6-2x) + (1-x) - 2x = 7 - 5x. 



If 1 < x < 3,    then 6 - 2x > 0 and hence |6-2x} = (6-2x);

                      x - 1 > 0  and hence |x-1| = (x-1);

                      thus the entire function is f(x) = (6-2x) + (x-1) - 2x = 5 - 3x.



Lastly, if x > 3 then 6 - 2x < 0 and hence |6-2x} = (-6+2x);

                      x - 1 > 0  and hence |x-1| = (x-1);

                      thus the entire function is f(x) = (-6+2x) + (x-1) - 2x = -7 + x.


So you have the expressions for f(x) piecewise linear and can draw it like this





Plot y = |6-2x|+|X-1|-2X