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Question 1090462: If f(3)=1 and f(2x)=2f(x) for all x, find f^{-1}(64).
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
The goal is to use the definition f(2x) = 2*f(x) to build up to a case where f(x) = 64. If we know that x value, then we can find the inverse result.
f(3) = 1
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f(2x) = 2*f(x)
f(2*3) = 2*f(3) ... replace every x with 3
f(6) = 2*1 ... replace f(3) with 1
f(6) = 2
-----------------
f(2x) = 2*f(x)
f(2*6) = 2*f(6) ... replace every x with 6
f(12) = 2*2 ... replace f(6) with 2
f(12) = 4
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f(2x) = 2*f(x)
f(2*12) = 2*f(12) ... replace every x with 12
f(24) = 2*4 ... replace f(12) with 4
f(24) = 8
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f(2x) = 2*f(x)
f(2*24) = 2*f(24) ... replace every x with 24
f(48) = 2*8 ... replace f(24) with 8
f(48) = 16
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f(2x) = 2*f(x)
f(2*48) = 2*f(48) ... replace every x with 48
f(96) = 2*16 ... replace f(48) with 16
f(96) = 32
-----------------
f(2x) = 2*f(x)
f(2*96) = 2*f(96) ... replace every x with 96
f(192) = 2*32 ... replace f(96) with 32
f(192) = 64
-----------------
Now we can apply the inverse
f(192) = 64
f^{-1}[ f(192) ] = f^{-1}[ 64 ]
192 = f^{-1}[ 64 ]
f^{-1}[ 64 ] = 192
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