Let $$f(x)$$ be defined for all x>$$0$$ and be continuous. Let $$f(x)$$ satisfies $$fxy=f(x)$$−$$f(y)$$ for all x,y and $$f(e)=$$ $$1$$. Then

Correct option is 4f(x)=loge⁡x

Questions

$Let f (x) be defined for all x > 0 and be continuous. Let$ $f (x) satisfies f (\frac{x}{y}) = f (x) - f (y) for all x, y and f (e) = 1 . T h e n$

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f(x) is bounded

f1x→0 as x→0

xf(x)→1 as x→0

f(x)=loge⁡x

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detailed solution

Correct option is D

f(x) is continuous for all x>0 and fxy=f(x)−f(y) Also, f(e)=1 Therefore, clearly, f(x)=loge⁡x satisfies all these properties. Thus, f(x)=loge⁡x, which is an unbounded function.

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Let f(x) be defined for all x>0 and be continuous. Let f(x) satisfies fxy=f(x)−f(y) for all x,y and f(e)= 1. Then

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

$Let f (x) be defined for all x > 0 and be continuous. Let$ $f (x) satisfies f (\frac{x}{y}) = f (x) - f (y) for all x, y and f (e) = 1 . T h e n$