Question 891793
<pre>

(fog)(x)=x²-2x+1 factors as (x-1)², which is also (-x+1)²

The easiest pair of functions is this one

f(x) = x², g(x) = x-1 

So choose f(x) = (ax+b)² and g(x) = cx+d

Then fog(x) = [a(cx+d)+b]²

Then fog(x) = [acx+ad+b]²    

You want that to come out to be either (x-1)² or (-x+1)²

1. Make up any number for a
2. Then c will have to be either the reciprocal of a or the
negative of the reciprocal of a so we'll get x if we're 
making (x-1)², or -x if we're making (-x+1)²
3, Then we choose b to make ad+b come out to be -1 if we're
making (x-1)² or +1 if we're making (-x+1)².
 
Here are 6 made up that way.  We can make up as many as we like.

{{{"f(x)"}}}{{{""=""}}}{{{(4x+9)^2}}},{{{"g(x)"}}}{{{""=""}}}{{{expr(-1/4)x-2}}}
{{{fog(x)}}}{{{""=""}}}{{{(4(expr(-1/4)x-2)+9)^2}}} 
{{{fog(x)}}}{{{""=""}}}{{{(-x-8+9)^2}}}
{{{fog(x)}}}{{{""=""}}}{{{(-x+1)^2}}}
{{{fog(x)}}}{{{""=""}}}{{{x^2-2x+1}}}

----------------------------

{{{"f(x)"}}}{{{""=""}}}{{{(5x-6)^2}}},{{{"g(x)"}}}{{{""=""}}}{{{expr(1/5)x+1}}}
{{{fog(x)}}}{{{""=""}}}{{{(5(expr(1/5)x+1)-6)^2}}} 
{{{fog(x)}}}{{{""=""}}}{{{(x+5-6)^2}}}
{{{fog(x)}}}{{{""=""}}}{{{(x-1)^2}}}
{{{fog(x)}}}{{{""=""}}}{{{x^2-2x+1}}}

----------------------------

{{{"f(x)"}}}{{{""=""}}}{{{(2x-5)^2}}},{{{"g(x)"}}}{{{""=""}}}{{{expr(-1/2)x+3}}}
{{{fog(x)}}}{{{""=""}}}{{{(2*(expr(-1/2)x+3)-5)^2}}} 
{{{fog(x)}}}{{{""=""}}}{{{(-x+6-5)^2}}}
{{{fog(x)}}}{{{""=""}}}{{{(-x+1)^2}}}
{{{fog(x)}}}{{{""=""}}}{{{x^2-2x+1}}}

----------------------------

{{{"f(x)"}}}{{{""=""}}}{{{(2x-1)^2}}},{{{"g(x)"}}}{{{""=""}}}{{{expr(1/2)x}}}
{{{fog(x)}}}{{{""=""}}}{{{(2*expr(1/2)x-1)^2}}} 
{{{fog(x)}}}{{{""=""}}}{{{(x-1)^2}}}
{{{fog(x)}}}{{{""=""}}}{{{x^2-2x+1}}}

----------------------------

{{{"f(x)"}}}{{{""=""}}}{{{(x+1)^2}}},{{{"g(x)"}}}{{{""=""}}}{{{-x}}}
 {{{fog(x)}}}{{{""=""}}}{{{((-x)+1)^2}}}
{{{fog(x)}}}{{{""=""}}}{{{(-x+1)^2}}}
{{{fog(x)}}}{{{""=""}}}{{{x^2-2x+1}}}

----------------------------

{{{"f(x)"}}}{{{""=""}}}{{{(3x+2)^2}}},{{{"g(x)"}}}{{{""=""}}}{{{expr(1/3)x-1}}}
{{{fog(x)}}}{{{""=""}}}{{{(3(expr(1/3)x-6)+9)^2}}} 
{{{fog(x)}}}{{{""=""}}}{{{(x-6+5)^2}}}
{{{fog(x)}}}{{{""=""}}}{{{(x-1)^2}}}
{{{fog(x)}}}{{{""=""}}}{{{x^2-2x+1}}}

Edwin</pre>