# Thread: Find fog and its domain and range?

1. ## 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).

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

2. 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??

3. 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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### domain and range of Fog

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