Acertain function f satisfies the equation f(x)+2*f(6-x) = x for all real numbers x. The value of f(1) isa)3b)5c)0d)-3Correct answer is option 'A'. Can you explain this answer?

Question

Answer

Solution:

Given,

f(x) * 2*f(6-x) = x

Let's put x = 1,

f(1) * 2*f(5) = 1

Now, let's put x = 5,

f(5) * 2*f(1) = 5

Multiplying both equations, we get,

[f(1) * 2*f(5)] * [f(5) * 2*f(1)] = 1 * 5

4*f(1)^2 * f(5)^2 = 5

f(1)^2 * f(5)^2 = 5/4

Now, let's put x = 4,

f(4) * 2*f(2) = 4

f(2) = 2*f(4)/4 = f(4)/2

Now, let's put x = 2,

f(2) * 2*f(4) = 2

f(4) = f(2)*2/4 = f(2)/2

Substituting f(4) and f(2) in terms of f(1), we get,

f(5) = f(1)/2

f(4) = f(1)/4

Substituting these values in f(1)^2 * f(5)^2 = 5/4, we get,

f(1)^2 * (f(1)/2)^2 = 5/4

f(1)^4 = 5/4 * 4 = 5

Taking square root,

f(1)^2 = √5

Since f(1) is a real number,

f(1) = ±√5

But f(1) cannot be negative, as f(x) is a function.

Therefore,

f(1) = √5

Hence, the correct option is A.

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