Question

Write explicitly, functions of $y$ defined by the following equations and also find the domains of definition of the given implicit functions:

$x+\mid y\mid =2y$

• A. $y=\{\begin{array}{c}xx<0\\ \frac{x}{3}x\ge 0\end{array}$
• B. $y=\{\begin{array}{c}\frac{x}{3}x<0\\ xx\ge 0\end{array}$
• C. $y=\{\begin{array}{c}\frac{x}{3}x<0\\ 3xx\ge 0\end{array}$
• D. None of these

Answer 1

The correct option is B $y=\{\begin{array}{c}\frac{x}{3}x<0\\ xx\ge 0\end{array}$

$x+\mid y\mid =2y$
$|y|=-y$ when $y<0$
$\Rightarrow x-y=2y$
$\Rightarrow y=\frac{x}{3}$
$|y|=y$ when $y\ge 0$
$\Rightarrow x+y=2y$
$\Rightarrow y=x$
Clearly, $x$ varies as $y$.
$y=\{\begin{array}{cc}\frac{x}{3}& x<0\\ x& x\ge 0\end{array}$