The sum of all solutions to f(f(f(x))) is 3 .
The graph of y=f(x) is a parabola, and the graph of y=f(f(x)) is the parabola reflected across the x-axis. Let A, B, and C be the vertices of the graph of y=f(x) as shown below.
The graphs of y=f(x) and y=−3 intersect at points B and C, so f(B)=f(C)=−3. Since B and C are symmetric with respect to the y-axis, B and C must be the vertices of y=f(f(x)).
Similarly, if D is the vertex of y=f(f(f(x))), then D is also the vertex of y=f(f(x)). Thus, D is the point where the graph of y=f(f(x)) intersects the vertical line through (3,−1).
Note that the graphs of y=f(x) and y=f(f(x)) intersect at (3,−1), so f(3)=f(f(3))=−1.
Since f(B)=−3 and 3 is between B and C, it follows that f′(3) is positive.
Hence, D is above point B, which means that f(f(f(3)))=−3 has only one solution, x=3.
Therefore, the sum of all solutions to f(f(f(x))) is 3 .
Question
A portion of the graph of y = f(x) is shown in red below, where f(x) is a quadratic function. The distance between grid lines is 1 unit. What is the sum of all distinct numbers x such that f(f(f(x)))=-3 ?