Thread: Find fog and its domain and range?

1. 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

2. 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??

3. 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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domain and range of Fog

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