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

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

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Answer 1 · 2017-03-04T09:14:03

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Explanation:

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

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

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

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

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

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

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

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

Adding $6$ $6$ to both ends we get:

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

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

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

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

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

1 Answer

Explanation:

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