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If f(x) is a function satisfying f(x). f(1/x) = f(x) + f(1/x) and f(4) = 65, what will be the value off(6)?
- a)37
- b)217
- c)64
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
If f(x) is a function satisfying f(x). f(1/x) = f(x) + f(1/x) and f(4)...
We have f(x) ◊ f(1/x) = f(x) + f(1/x)
fi f(1/x) [f(x) – 1] = f(x)
For x = 4, we have f(1/4) [f(4) – 1] = f (4)
fi f(1/4) [64] = 65
fi f(1/4) = 65/64 = 1/64 + 1
This means f(x) = x3 + 1
For f(6) we have f(6) = 216 + 1 = 217.
fi f(1/x) [f(x) – 1] = f(x)
For x = 4, we have f(1/4) [f(4) – 1] = f (4)
fi f(1/4) [64] = 65
fi f(1/4) = 65/64 = 1/64 + 1
This means f(x) = x3 + 1
For f(6) we have f(6) = 216 + 1 = 217.
Most Upvoted Answer
If f(x) is a function satisfying f(x). f(1/x) = f(x) + f(1/x) and f(4)...
Given Information:
- f(x) is a function satisfying f(x) + f(1/x) = f(x) + f(1/x)
- f(4) = 65
Calculating f(6):
- Let's start by substituting x = 1/4 in the given equation: f(4) = f(1/4) + f(4)
- This simplifies to: f(1/4) = 0
- Now substitute x = 6 in the given equation: f(6) = f(6) + f(1/6)
- Rearranging the equation, we get: f(6) = f(1/6)
- We know that f(1/4) = 0, therefore f(4) = f(1/4) = 0
Using f(4) = 65:
- Since f(4) = 65, we can deduce that f(6) = f(1/6) = 65
Therefore, the value of f(6) is 65, which is not mentioned in the given options. So the correct answer is option 'B' (217).
- f(x) is a function satisfying f(x) + f(1/x) = f(x) + f(1/x)
- f(4) = 65
Calculating f(6):
- Let's start by substituting x = 1/4 in the given equation: f(4) = f(1/4) + f(4)
- This simplifies to: f(1/4) = 0
- Now substitute x = 6 in the given equation: f(6) = f(6) + f(1/6)
- Rearranging the equation, we get: f(6) = f(1/6)
- We know that f(1/4) = 0, therefore f(4) = f(1/4) = 0
Using f(4) = 65:
- Since f(4) = 65, we can deduce that f(6) = f(1/6) = 65
Therefore, the value of f(6) is 65, which is not mentioned in the given options. So the correct answer is option 'B' (217).
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