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Show the signum function f:RrarrR given ...

Show the signum function `f:RrarrR` given by `f(x)={(1" if "xgt0),(0" if "x=0 " is"),(-1" if " xlt0):}`neither one-one nor onto

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Explore conceptually related problems

Show that the signum function f : R to R given by f(x) = {(1, if, x gt 0),(0, if, x = 0),(-1, if, x lt 0):} is neither one-one nor onto.

Show that the function f given by f(x)= {{:(x^2+2," if "x ne 0),(1," if "x=0):} is not continuous at x=0.

Knowledge Check

  • If y={:{(x","xgt0),(0","x=0","),(-x","xlt0):} then at x=0 , y is

    A
    not continuous
    B
    continuous but not differentiable
    C
    differentiable
    D
    None of these.

  • The inverse of the function f: R rarr R given by f(x) = log_{a}(x+sqrt(x^2+1)(a gt 0, a ne 1) is

    A
    `1/2(a^x+a^-x)`
    B
    `1/2(a^x-a^-x)`
    C
    `1/2((a^x+a^-x)/(a^x-a^-x))`
    D
    `1/2((a^x-a^-x)/(a^x+a^-x))`

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    Discuss the continuity of the function f given by {{:(x," if "x ge 0),(x^3," if "x lt 0):} .

    Discuss the continuity of the function f, where f is defined by f(x)={{:(2x," if "x lt 0),(0," if "0 le x le 1),(4x," if "x ge 1):} .

    Let f : R to R be the Signum Function defined as f (x) = {{:(1",", x gt 0), ( 0"," , x =0), ( -1 "," , x lt 0):} and g : R to R be the Greatest Integer Function given by g (x) = [x], where [x] is greatest integer less than or equal to x. Then does fog and gof coincide in (0,1] ?

    Show that the function f: N to N given by f(x)=2x is one-one but not onto.

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