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Math Help - Find fog and its domain and range?

  1. #1
    Super Member fardeen_gen's Avatar
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    Find fog and its domain and range?

    Find f(g(x)) if f(x) = [x] + \{x\}^2 and g(x) = [x] + \sqrt{\{x\}} and also find the domain and range of f(x).

    Answer:
    Spoiler:
    f\circ g = x,\ f\circ g:\mathbb{R}\rightarrow \mathbb{R}


    I am not getting a matching answer. Anyone?

    NOTE: [x] is floor function/greatest integer function
    {x} is fractional part function
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  2. #2
    AMI
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    If you have two numbers a\in\mathbb{Z} and b\in[0,1), then [a+b]=a and \{a+b\}=b. With these you immediately get (f\circ g)(x)=x,\forall x.
    And then you need the domain and range of f, like you wrote in the exercise or that of f\circ g, as you wrote in the answer??
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  3. #3
    Moo
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    Hello,

    g(x)=[x]+\sqrt{\{x\}}
    \{x\}\in [0,1) \Rightarrow \sqrt{\{x\}}\in [0,1)

    Then [g(x)]=[x] and \{g(x)\}=\sqrt{\{x\}}

    Hence f(g(x))=[x]+\{x\}=x

    As for the range, there is no problem with the square root, since the fractional part is always positive. So it's from \mathbb{R}\to\mathbb{R}
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