+23246 For -4 ≤ x ≤ -1 ---> f(x) = -(3/4)x + 1/4 ---> f( f(x) ) = -(3/4)( -(3/4)x + 1/4) + 1/4 = (9/16)x + 1/16
---> 3 = (9/16)x + 1/16 ---> x = 47/9 (not in the domain)
For -1 ≤ x ≤ 3 ---> f(x) = x + 2 ---> f( f(x) ) = (x + 2) + 2 = x + 4
---> 3 = x + 4 ---> x = -1 ---> one point
For 3 ≤ x ---> f(x) = -x + 8 ---> f( f(x) ) = -(-x + 8) + 8 = x - 8
---> 3 = x - 8 ---> x = 11 ---> one point
![The graph of the function $f(x)$ is shown below. How many values of $x$ satisfy $f(f(x)) = 3$? [asy] import graph; size(7.4cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.4,xmax=5.66,ymin=-1.05,ymax=6.16; Label laxis; laxis.p=fontsize(10); xaxis("$x$",-4.36,5.56,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,OmitTick(0)),Arrows(6),above=true); yaxis("$y$",-0.92,6.12,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,OmitTick(0)),Arrows(6),above=true); draw((xmin,(-(0)-(-2)*xmin)/-2)--(-1,(-(0)-(-2)*-1)/-2),linewidth(1.2)); draw((-1,1)--(3,5),linewidth(1.2)); draw((3,(-(-16)-(2)*3)/2)--(xmax,(-(-16)-(2)*xmax)/2),linewidth(1.2)); // draw((min,(-(-9)-(0)*xmin)/3)--(xmax,(-(-9)-(0)*xmax)/3),linetype("6pt 6pt")); label("$f(x)$",(-3.52,4.6),SE*lsf); //dot((-1,1),ds); dot((3,5),ds); dot((-3,3),ds); dot((1,3),ds); dot((5,3),ds); dot((-4.32,4.32),ds); dot((5.56,2.44),ds); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]](/api/ssl-img-proxy?src=https%3A%2F%2Fdata.artofproblemsolving.com%2Faops20%2Flatex%2Fimages%2F65cd565c391a45c768dde930e2b6e1b8899086b1.png)
