Question

Write the domain and the range of the function!!

• $\sf f(x) = \displaystyle \sqrt{x - |x| }$

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Answer 1

We're asked to find the domain and range of the function,

$\longrightarrow f(x)=\sqrt{x-|x|}$

In other way,

$\longrightarrow f(x)=\left\{\begin{array}{lr}\sqrt{2x},&x\leq0\\0,&x\geq0\end{array}\right.$

Consider,

$\longrightarrow f(x)=\sqrt{2x},\quad\!\!x\in(-\infty,\ 0]$

But, since $\sqrt x$ is defined only for $x\in[0,\ \infty),\ \sqrt{2x}$ is only defined for,

$\longrightarrow 2x\in[0,\ \infty)$

$\longrightarrow x\in[0,\ \infty)$

Then, $f(x)=\sqrt{2x}$ only for,

$\longrightarrow x\in(-\infty,\ 0]\cap[0,\ \infty)$

$\longrightarrow x\in\{0\}\quad\quad\dots(1)$

And so,

$\longrightarrow f(0)=\sqrt{2\cdot0}$

$\longrightarrow f(0)=0$

$\Longrightarrow f(x)\in\{0\}\quad\quad\dots(2)$

Consider,

$\longrightarrow f(x)=0,\quad\!\!x\in[0,\ \infty)$

Here the function behaves like a constant function and there's no restriction, for every,

$\longrightarrow x\in[0,\ \infty)\quad\quad\dots(3)$

The function has the value 0 only here, i.e.,

$\longrightarrow f(x)\in\{0\}\quad\quad\dots(4)$

Now taking (1) ∨ (3),

$\longrightarrow x\in\{0\}\cup[0,\ \infty)$

$\longrightarrow\underline{\underline{x\in[0,\ \infty)}}$

This is the domain of the function.

And taking (2) ∨ (4),

$\longrightarrow f(x)\in\{0\}\cup\{0\}$

$\longrightarrow\underline{\underline{f(x)\in\{0\}}}$

This is the range of the function.