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An even polynomial function `f(x)`satisfies a relation `f(2x)(1-f(1/(2x)))+f*16^2y)=f(-2)-f(4x y)AAx ,y in R-{0}a n df(4)=-255 ,f(0)=1.`Then the value of `|(f(2)+1)//2|`is_________ |
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Answer» Correct Answer - -15 We have `f(2x)-f(2x) f((1)/(2x))+f(16x^(2)y)=f(-2)-f(4xy)` Replacing y by `(1)/(8x^(2)),` we get `f(2x)-f(2x)((1)/(2x))+f(2)=f(-2)-f((1)/(2x))` ` :. f(2x)+f((1)/(2x))=f(2x)f((1)/(2x)) ` [As `f(x)` is even] ` :. f(2x)=1+-(2x)^(n)` `or f(x)=1+-x^(n)` Now, `f(4)=1+-4^(n)= -255 " "` (Given) Taking negative sign, we get `256=4^(n) or n=4.` Hence, `f(x)=1-x^(4)`, which is an even function. Therefore, `f(2)= -15.` |
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