# Showing that relation $f$ is a function

How can I determine if the relation $f$ on $\mathbb{R}$ given by $xfy\Leftrightarrow (y(2x-3)-3x=y(x^2-2x)-5x^2)$ is a function?

I've tried plotting the function as a quadratic function, and doing the vertical line test (but without success)

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Unless you made a mistake, the condition is $$y=\frac{3x-5x^2}{2x-3-x^2+2x}=\frac{3x-5x^2}{-x^2+4x-3}.$$ Of course as long as $-x^2+4x-3 \neq 0$.
That is, $x\ne 3,1$. So, it is not a function on the whole domain $\Bbb R$, but, a function from $\Bbb R\setminus \{1,3\}$. For a final answer, try more $y$ values with $x=1$ that satisfy the original equaiton. –  Berci Oct 24 '12 at 11:26