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Which of the following statements are not correct?
1- y as a function of $x$ is not defined for all real $x$.
2- y as a function of $x$ is not continuous at $x=0$.
3- y as a function of $\mathrm{x}$ is differentiable for all $\mathrm{x}$. Select the correct answer using the code given below.
Consider the equation $\mathrm{x}+|\mathrm{y}|=2 \mathrm{y}$
Options:
1- y as a function of $x$ is not defined for all real $x$.
2- y as a function of $x$ is not continuous at $x=0$.
3- y as a function of $\mathrm{x}$ is differentiable for all $\mathrm{x}$. Select the correct answer using the code given below.
Consider the equation $\mathrm{x}+|\mathrm{y}|=2 \mathrm{y}$
Solution:
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Verified Answer
The correct answer is:
1,2 and 3
$x+|y|=2 y$
$x=2 y-|y|$
$2 y-|y|=x$
$2 y-y=x \quad[$ for $y \geq 0]$
$y=x$
$2 y+y=x \quad[$ for $y < 0]$
$3 y=x$
$y=\frac{1}{3} x$
$y=\left\{\begin{array}{ll}x & y \geq 0 \\ \frac{1}{3} x & y < 0\end{array}\right.$ function is defined for all value of $\mathrm{x}$.
or $\mathrm{y}=\left\{\begin{array}{ll}x & ; & x \geq 0 \\ \frac{1}{3} x & ; & x < 0\end{array}\right.$
$\therefore$ by checking $y$ as a function of $x$ is continuous at $x=0$, but not differentiable at $x=0$. So all of the statements are not correct.
$x=2 y-|y|$
$2 y-|y|=x$
$2 y-y=x \quad[$ for $y \geq 0]$
$y=x$
$2 y+y=x \quad[$ for $y < 0]$
$3 y=x$
$y=\frac{1}{3} x$
$y=\left\{\begin{array}{ll}x & y \geq 0 \\ \frac{1}{3} x & y < 0\end{array}\right.$ function is defined for all value of $\mathrm{x}$.
or $\mathrm{y}=\left\{\begin{array}{ll}x & ; & x \geq 0 \\ \frac{1}{3} x & ; & x < 0\end{array}\right.$
$\therefore$ by checking $y$ as a function of $x$ is continuous at $x=0$, but not differentiable at $x=0$. So all of the statements are not correct.
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