# Do the following equations define functions: (i) y = x^2-5x (ii) x = y^2-5y ?

Mar 4, 2017

See explanation...

#### Explanation:

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First equation: $y = {x}^{2} - 5 x$

For the first equation, putting $x = - 6$ we find:

$y = {x}^{2} - 5 x = {\left(- 6\right)}^{2} - 5 \left(- 6\right) = 36 + 30 = 66$

Note that the value of $y$ is uniquely determined by the value of $x$. This is true for any value of $x$, so $y$ is a function of $x$.

graph{(y - x^2+5x)(x+6+0.0001y) = 0 [-11, 11, -11, 102]}

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Second equation: $x = {y}^{2} - 5 y$

For the second equation, putting $x = - 6$ we find:

$- 6 = x = {y}^{2} - 5 y$

Adding $6$ to both ends we get:

$0 = {y}^{2} - 5 y + 6 = \left(y - 2\right) \left(y - 3\right)$

So $y = 2$ or $y = 3$

Note that $y$ is not uniquely determined by the value of $x$, so is not a function of $x$.

graph{(x - y^2+5y)(x+6+0.0001y) = 0 [-12, 2, -2.5, 6]}