FunctionHard

Question

If f(x) = sgn , where [.] is the greatest integer function, then which of the following statement is/are true ?

Options

A.f(x) = 1
B.Range of f(x) contains only one integer
C.f(x) is periodic function.
D.The equation f(x) = 1 + x2 dt has infinitely many non-integral roots.

Solution

    
f(x) = sgn.
domain of the function is x ∈ [2n, 2n+1) where
n ∈ I. and tan = 0 ⇒ ∀ x ∈ [2n,2n + 1)
and sgn = 1 for x in domain
∴ f(x) = 1 ∀ x ∈ [2n, 2n + 1) graph of y = f(x)
Also, f(x) = 1 + x2dt = 1 + x2 - 1
⇒  x2 = 0
⇒  x = 0 only one root.

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