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Let \(g(x)\) be a function piecewise defined as \(g(x) = \left\{ \begin{array}{cl} -x & x\le 0, \\ 2x-41 & x>0. \end{array} \right.\)If  \(a\)is negative, find \(a\) so that \(g(g(g(10.5)))=g(g(g(a)))\) .

 Apr 26, 2019
 #1
avatar+27844 
+4

 a = -30.5:

 

g(10.5) = -20; g(-20) = 20; g(20) = -1 

 

 

a = -30.5:  g(-30.5) = 30.5;  g(30.5) = 20;  g(20) = -1

 

Edited to correct silly mistake!  Thanks for pointing this out Melody.

 Apr 27, 2019
edited by Alan  Apr 27, 2019
edited by Alan  Apr 28, 2019
 #2
avatar+100814 
+2

Thanks Alan laugh

I find these difficult to get my head around too.

I find it easier if I draw a graph and I can see what I am doing visually.

 

g(10.5)=-20

g(-20)=20

g(20)=-1

          Now going backwards.

         g(what)=20    there are two answers, -20 and 30.5   (this is easy to see from the graph)

          so

          g(-20)=20      and     g(30.5)=20

                   now go backwards again

                   g(what)=-20      and     g(what)=30.5  looking at the graph I can see that

                   g(10.5)=-20      and      g(-30.5)=30.5    and   g(35.75)=30.5

 

so

g(10.5) is what we started with

You are told that a is negative so  35.75 is no good

a=-30.5

 

check

g(-30.5)=30.5

g(30.5)=20

g(20)=-1  = g(g(g(10.5)))

ie

g(g(g(-30.5)=g(g(g(10.5)))

a=-30.5

 

 Apr 27, 2019

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