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A function ๐‘“(๐‘ฅ) has the property that, for all positive ๐‘ฅ, 3๐‘“(๐‘ฅ)+7๐‘“(2016๐‘ฅ)=2๐‘ฅ.What is the value of ๐‘“(8)?

Question

A function f(x) f(x) has the property that, for all positive x x ,

3f(x)+7f(2016x)=2x. 3f(x) + 7f(2016x) = 2x.
What is the value of f(8) f(8) ?

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Solution

To solve this problem, we need to find a pattern or relationship that can help us determine the value of f(8).

  1. Let's start by setting x = 1/2016 in the given equation. We get:

    3f(1/2016) + 7f(1) = 2/2016

  2. Now, let's set x = 1 in the given equation. We get:

    3f(1) + 7f(2016) = 2

  3. From the first equation, we can express f(1) as:

    f(1) = (2/2016 - 3f(1/2016)) / 7

  4. Substitute f(1) from equation 3 into equation 2:

    3((2/2016 - 3f(1/2016)) / 7) + 7f(2016) = 2

  5. Simplify the equation to find the value of f(2016):

    f(2016) = (2 - 3(2/2016 - 3f(1/2016)) / 7) / 7

  6. Now, let's set x = 8 in the given equation. We get:

    3f(8) + 7f(20168) = 28

  7. Substitute f(2016*8) from equation 5 into equation 6:

    3f(8) + 7((2 - 3(2/2016 - 3f(1/2016)) / 7) / 7) = 16

  8. Solve the equation to find the value of f(8).

Please note that this is a complex problem and it may require additional steps or a different approach depending on the specific properties of the function f(x).

This problem has been solved

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