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 $f(f(f(x)))=-3$ ?

[asy]

size(150);

real ticklen=3;

real tickspace=2;

real ticklength=0.1cm;

real axisarrowsize=0.14cm;

pen axispen=black+1.3bp;

real vectorarrowsize=0.2cm;

real tickdown=-0.5;

real tickdownlength=-0.15inch;

real tickdownbase=0.3;

real wholetickdown=tickdown;

void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) {

import graph;

real i;

if(complexplane) {

label("$\textnormal{Re}$",(xright,0),SE);

label("$\textnormal{Im}$",(0,ytop),NW);

} else {

label("$x$",(xright+0.4,-0.5));

label("$y$",(-0.5,ytop+0.2));

}

ylimits(ybottom,ytop);

xlimits( xleft, xright);

real[] TicksArrx,TicksArry;

for(i=xleft+xstep; i<xright; i+=xstep) {

if(abs(i) >0.1) {

TicksArrx.push(i);

}

}

for(i=ybottom+ystep; i<ytop; i+=ystep) {

if(abs(i) >0.1) {

TicksArry.push(i);

}

}

if(usegrid) {

xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.22),extend=true),p=invisible);//,above=true);

yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.22),extend=true), p=invisible);//,Arrows);

}

if(useticks) {

xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));

yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));

} else {

xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));

yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));

}

};

rr_cartesian_axes(-8,4,-6,6);

real f(real x) {return x^2/4+x-3;}

draw(graph(f,-8,4,operator ..), red);

[/asy]
First, we note that there are two points on the graph whose $y$-coordinates are $-3$. These are $(-4,-3)$ and $(0,-3)$. Therefore, if $f(f(f(x)))=-3$, then $f(f(x))$ equals $-4$ or $0$.

There are three points on the graph whose $y$-coordinates are $-4$ or $0$. These are $(-2,-4),$ $(-6,0),$ and $(2,0)$. Therefore, if $f(f(x))$ is $-4$ or $0$, then $f(x)$ equals $-2,$ $-6,$ or $2$.

There are four points on the graph whose $y$-coordinates are $-2$ or $2$ (and none whose $y$-coordinate is $-6$). The $x$-coordinates of these points are not integers, but we can use the symmetry of the graph (with respect to the vertical line $x=-2$) to deduce that if these points are $(x_1,-2),$ $(x_2,-2),$ $(x_3,2),$ and $(x_4,2),$ then $x_1+x_2=-4$ and $x_3+x_4=-4$. Therefore, the sum of all four $x$-coordinates is $\boxed{-8}$.
-8
Level 5
Algebra
Problem: 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 $f(f(f(x)))=-3$ ?

[asy]

size(150);

real ticklen=3;

real tickspace=2;

real ticklength=0.1cm;

real axisarrowsize=0.14cm;

pen axispen=black+1.3bp;

real vectorarrowsize=0.2cm;

real tickdown=-0.5;

real tickdownlength=-0.15inch;

real tickdownbase=0.3;

real wholetickdown=tickdown;

void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) {

import graph;

real i;

if(complexplane) {

label("$\textnormal{Re}$",(xright,0),SE);

label("$\textnormal{Im}$",(0,ytop),NW);

} else {

label("$x$",(xright+0.4,-0.5));

label("$y$",(-0.5,ytop+0.2));

}

ylimits(ybottom,ytop);

xlimits( xleft, xright);

real[] TicksArrx,TicksArry;

for(i=xleft+xstep; i<xright; i+=xstep) {

if(abs(i) >0.1) {

TicksArrx.push(i);

}

}

for(i=ybottom+ystep; i<ytop; i+=ystep) {

if(abs(i) >0.1) {

TicksArry.push(i);

}

}

if(usegrid) {

xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.22),extend=true),p=invisible);//,above=true);

yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.22),extend=true), p=invisible);//,Arrows);

}

if(useticks) {

xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));

yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));

} else {

xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));

yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));

}

};

rr_cartesian_axes(-8,4,-6,6);

real f(real x) {return x^2/4+x-3;}

draw(graph(f,-8,4,operator ..), red);

[/asy]
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To solve for the sum of all distinct numbers \( x \) such that \( f(f(f(x))) = -3 \), we first need to determine the function \( f(x) \) from the graph. The graph appears to be a quadratic function, which can be expressed in the standard form \( f(x) = ax^2 + bx + c \). By analyzing the vertex and intercepts, we can deduce that \( f(x) = \frac{x^2}{4} + x - 3 \).

Next, we need to find the values of \( x \) for which \( f(x) = -3 \). Setting the equation \( \frac{x^2}{4} + x - 3 = -3 \) simplifies to \( \frac{x^2}{4} + x = 0 \). Factoring out \( x \) gives \( x(\frac{x}{4} + 1) = 0 \), leading to the solutions \( x = 0 \) and \( x = -4 \).

Next, we need to find \( f(f(x)) \) and set it equal to the previously found values of \( x \). We calculate \( f(0) \) and \( f(-4) \). For \( f(0) \), we have \( f(0) = -3 \). For \( f(-4) \), we compute \( f(-4) = \frac{(-4)^2}{4} + (-4) - 3 = 4 - 4 - 3 = -3 \). Thus, both \( f(0) \) and \( f(-4) \) yield \( -3 \).

Now we need to find \( f(f(f(x))) = -3 \). Since \( f(f(x)) \) can yield \( -3 \) from both \( 0 \) and \( -4 \), we need to find \( f(x) = 0 \) and \( f(x) = -4 \).

Setting \( f(x) = 0 \) gives \( \frac{x^2}{4} + x - 3 = 0 \). Multiplying through by 4 results in \( x^2 + 4x - 12 = 0 \). Using the quadratic formula, \( x = \frac{-4 \pm \sqrt{16 + 48}}{2} = \frac{-4 \pm 8}{2} \), leading to \( x = 2 \) and \( x = -6 \).

Next, we set \( f(x) = -4 \). This gives \( \frac{x^2}{4} + x - 3 = -4 \), simplifying to \( \frac{x^2}{4} + x + 1 = 0 \). Multiplying through by 4 results in \( x^2 + 4x + 4 = 0 \), which factors to \( (x + 2)^2 = 0 \), yielding \( x = -2 \).

Now we have the distinct values \( x = 2, -6, -2 \). The sum of these values is \( 2 + (-6) + (-2) = -6 \).

Finally, we need to consider the values of \( f(f(f(x))) \) that yield \( -3 \). The distinct values \( 0 \) and \( -4 \) lead to the final distinct values \( 2, -6, -2 \). The sum of all distinct numbers \( x \) such that \( f(f(f(x))) = -3 \) is \( -8 \).<|END_OF_LATENT|> The final answer is $-8$.