Question

g(x) = 1 + x - [x] : R -> [1,2)

f(x) = sgn. x : R -> { -1, 0, 1}

Find f[g(x)]

Note :-

[x] denotes Greatest Integer Function

sgn x denotes signum function.


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Answer 1

Answer:

Step-by-step explanation:

let x be an integer

             then  [x] = x

             then g(x) = 1+x - [x] = 1 + x - x  = 1  

                     f(g(x)) =f(1)  = 1  

let x be a non integer  

           let x = n +r  here  0 ≤ r < 1   and n is integer

             then     [x] = n

then g(x) = 1+x - [x] = 1 + n+r  - n  = 1 +r  

            f(g(x)) =f(1+r )  = 1  

so  f[g(x)] = 1  : R ↔ [1,2)

NOTE

f[g(x)]    IS A CONSTANT FUNCTION  

Answer 2

$<font color = "blue"><marquee>Answer is proved </font color="blue"></marquee>$

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