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

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


    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 $\displaystyle a\in\mathbb{Z}$ and $\displaystyle b\in[0,1)$, then $\displaystyle [a+b]=a$ and $\displaystyle \{a+b\}=b$. With these you immediately get $\displaystyle (f\circ g)(x)=x,\forall x$.
    And then you need the domain and range of $\displaystyle f$, like you wrote in the exercise or that of $\displaystyle f\circ g$, as you wrote in the answer??
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  3. #3
    Moo
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    Hello,

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

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

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

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