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Suppose `f` is a real function satisfying `f(x+f(x))=4f(x)a n df(1)=4.` Then the value of `f(21)` is `16` `21` `64` `105`

A

16

B

64

C

4

D

44

Text Solution

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The correct Answer is:
To solve the problem, we need to find the value of \( f(21) \) given the functional equation \( f(x + f(x)) = 4f(x) \) and the condition \( f(1) = 4 \). ### Step-by-step solution: 1. **Start with the functional equation**: \[ f(x + f(x)) = 4f(x) \] 2. **Substitute \( x = 1 \) into the functional equation**: \[ f(1 + f(1)) = 4f(1) \] Since \( f(1) = 4 \), we have: \[ f(1 + 4) = 4 \cdot 4 \] This simplifies to: \[ f(5) = 16 \] 3. **Now substitute \( x = 5 \) into the functional equation**: \[ f(5 + f(5)) = 4f(5) \] We already found \( f(5) = 16 \), so: \[ f(5 + 16) = 4 \cdot 16 \] This simplifies to: \[ f(21) = 64 \] 4. **Conclusion**: The value of \( f(21) \) is \( 64 \). ### Final Answer: \[ f(21) = 64 \]
To solve the problem, we need to find the value of \( f(21) \) given the functional equation \( f(x + f(x)) = 4f(x) \) and the condition \( f(1) = 4 \). ### Step-by-step solution: 1. **Start with the functional equation**: \[ f(x + f(x)) = 4f(x) \] ...
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