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Suppose f is a real function satisfying f ( x + f ( x ) ) = 4 f ( x ) and f ( 1 ) = 4 . What is f ( 2 1 ) ?
The answer is 64.
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3 solutions
still i can't understand ... please give some explain why f(5)= 16??
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because f(1)=4, so f(1+f(1))=4f(1)=f(1+4)=f(5)=4f(1)=16
My name is NATNAEL SOLOMON. I am from Ethiopia. I am a grade 12 student. Let me explain you the question you are asking for (x+f(x))=4f(x) since f(1)=4 now substitute x=1 and f(x)=f(1)=4 so we have f(1+f(1))=f(1+4)=f(5)=4f(1)=4(4)=16. This is it. Thank you for your attention.Goodbye
Kinda tricky
that's crazy!
great.............
How f(21) - f(5 plus f(5))
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f(5)=16. so f(5+f(5))=f(5+16) cuz you sub in 16 for f(5) and then f(5+16)=f(21)
f ( x ) must be linear...
Let y = f ( x ) = a x + b
f ( x + y ) = a ( x + y ) + b = 4 y
Thus, a = 3 and b = 1
So, f ( x ) = 3 x + 1
f ( 2 1 ) = 2 1 ∗ 3 + 1 = 6 4
How to find b = 1 ?
Makes more sense than the previous ans...
Why must f(x) be linear? Looks intuitively correct, but any formal theorem/proof for this assertion?
f ( x ) is not necessarily a polynomial, and so, you can't say it is a linear by equating the degree on both sides like 1 2 = 1
Ez, the answer was in the statistics!
f ( x + f ( x ) ) = 4 f ( x ) = > f ( 1 + f ( 1 ) ) = 4 f ( 1 ) = 1 6
so f ( 5 ) = 1 6 .
Then f ( 2 1 ) = f ( 5 + f ( 5 ) ) = 4 f ( 5 ) = 6 4