The graph of the function f is shown below. How many solutions does the equation f(f(x)) = 6 have?

Note  that   f (-2)   =  6  and that f(2)  = 6

Which means that  f(x)  =  -2   or  f(x)  = 2

Note that  there is no point on the graph  where  f(x)  = -2    becaise  the graph is never negative

So....it must mean that  we are looking for   f (x)  = 2

Note that there  are three places on the graph where  f(x)   = 2   

There is actually  an algebraic way to show that there are three values which give us   f (f(x))  =  6

The slope of  the leftmost segment of the line  = (6 - 0) /(-2 - -4)  =  6/ 2  =  3

The equation of this segment  is   y = 3 ( x - - 4)  =  y = 3x + 12

So....we are looking  for the  x value where     f(x)  =2....so ...

  2 = 3x + 12  ...so   -10  = 3x  and x = -10/3

So    f (-10/3)  =  2

So   f ( f(x))  = f ( f(-10/3))  = f (2)  = 6

And the slope of the middle segment on the  graph   = (6-0)/(-2-0)  = -6/2  = -3

And the equation of this segment is  y = -3x

So....we are looking for the  x value where  f(x)  = 2  ...so

2  =  -3x ...  so     x = -2/3

So   f( f(x))  = f ( f(-2/3))  = f (2)  = 6

Finally......the slope of the leftmost segment is the same as the leftmost segment = 3

And the equation of this segment   is  y  =3x

So.....we are looking for the  x value  where f(x)  = 2....so

2 = 3x   ...so    x  = 2/3

So  f ( f(x))  = f(f(2/3) =  f(2)  = 6

And this confirms that   there are three solutions for  f (f(x))  =  6