One: From the meaning of the question, to make | x 2-4x+3 | = mx have four unequal real roots, an equation is needed.

X 2-4x+3 = MX and x2-4x+3 =-MX;

That is, x 2-(4+m) x+3 = 0, and x 2+(m-4) x+3 = 0.

There are two unequal real roots, so two inequalities can be obtained from the discriminant of the roots:

(m+4)^2- 12>； 0;

(m-4)^2- 12>； 0;

You can solve the solution set of two inequalities, and then you can find the intersection:

The first solution set is: {m | m & gt-4+2√3 or m.

The second solution set is: {m | m & gt4+√3 or m < 4-√3 }；;

Get the intersection: {m | m & gt4+√3 or (2 √ 3)-4 < m <; 4-√3 or m < -4-2√3}.

Second, the meaning of the problem: two functions that are inverse functions are symmetrical about Y=X;

Therefore, the image with Y=f(x) must pass (0,1);

Let t = 0.5x-1;

Then the function Y=f(t) must pass through the point (0,1);

Then 0.5x-1= 0;

So x = 2；;

So y=f(0.5x- 1) must pass (2,1);

So its inverse function image must pass through the point (1, 2).