Find fog and its domain and range?

**fardeen_gen** · June 25th 2009, 01:15 PM

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:

I am not getting a matching answer. Anyone?

NOTE: [x] is floor function/greatest integer function
{x} is fractional part function

**AMI** · June 26th 2009, 12:29 AM

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??

**Moo** · June 26th 2009, 01:08 AM

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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