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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]

Guest Nov 24, 2014

Best Answer 

 #3
avatar+91450 
+5

 

Here is the graph - 2 points

NOTE: I have not fully got my head around how to do this without graphing it.  

 

 

Melody  Nov 25, 2014
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3+0 Answers

 #1
avatar+17711 
+5

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

geno3141  Nov 24, 2014
 #2
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Sorry to say that this answer was wrong. Can you see if there was another answer to this possibly?

Guest Nov 24, 2014
 #3
avatar+91450 
+5
Best Answer

 

Here is the graph - 2 points

NOTE: I have not fully got my head around how to do this without graphing it.  

 

 

Melody  Nov 25, 2014

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