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