# What tis the complete domain D and range R of the following multivariable functions: w(x,y)=1/(x(y-1))

What tis the complete domain D and range R of the following multivariable functions: $w\left(x,y\right)=\frac{1}{x\left(y-1\right)}$
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$w\left(x,y\right)=\frac{1}{x\left(y-1\right)}$
denominator for rational function $\ne 0$
So, $x\left(y-1\right)\ne 0$
$x\ne 0\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\left(y-1\right)\ne 0$
$x\ne 0\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}y\ne 1$
Thus, $D={\mathbb{R}}^{2}-\left[0,1\right]$
The multivariable function
But oo is not included in the domain.
Thus, 0 is excluded from the range for $w\left(x,y\right).$
Hence , $R=\mathbb{R}-\left[0\right]$