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Most Important Selected Qs for JEE AdvancedMathematicsFunctions

Suppose that f: R R is a continuous function and satisfies the equation f(x) f(f(x))=1 for all x R . Further, if f(1000)=999 , then which of the following options are necessarily true? 1. f(500)= 1 500 2. f(199)= 1 199 3. f(2000)= 1 2000 4. f(235)= 1 235 5. f(1099)= 1 1099 6. f(x)= 1 x x R- 0,1000 7. No such function exists Enter the product of the number of all correct options. For example, if correct options are 2

Options

  1. A2
  2. B4
  3. C6
  4. D8

Correct answer

D. 8

Step-by-step solution

For all y in the range of f , we have y f(y)=1 , so f(y)= 1 y . Since 999 is in the range of f , we have f(999)= 1 999 , so 1 999 is in the range of f , then the range of f contains [ 1 999 , 999 ] by the intermediate value theorem, so for all y in [ 1 999 , 999 ], f(y)= 1 y . Hence statements 1, 2, 4 are all true. On the other hand, let y=g(x) be the equation of the line through the points (999, 1 999 ) and (1000,999) and considerd the function. f(x)= cases 999, & if x 1 999 1 x , & if 1 999 x 999 g(x), & if 999 x

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