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Function:

$f(x)= \frac{x-3}{x^2+2x-8} $

In terms of y:

$y= \frac{x-3}{x^2+2x-8} $

Then x isolated: $x= \frac{\sqrt{36y^2-16y+1}-2y+1}{2y} $

To find the Range we need to find the Domain of this 'new' Function.

1.- We must look if $\ \sqrt{36y^2-16y+1} $ is a Real number

2.- The Denominator $\ 2y $ should not be 0

1.- $\ \sqrt{36y^2-16y+1} $ will always be positive, so will be Real.

2.- We find the root: $\ 2y=0 $ so $\ y=0 $

Which left us that this Domain is all Real Numbers except 0, so this is the Range of the original Function.

But look at the graphic: Function

That wasn't the range, but i've found, i've found the roots of $\ {36y^2-16y+1} $ and those roots are that numbers accordingly to the graphic: .0752 and .03692, but i dont know how to argue this result.

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    I was looking for doing this way, i was wondering... thank you man2011-11-10

1 Answers 1

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The function inside the square root, $36y^2 - 16y + 1$, is less than zero on the interval $(\frac{1}{18}(4-\sqrt{7}),\frac{1}{18}(4+\sqrt{7})) = (0.075236,0.369208)$ which makes the function undefined on that interval. This appears to correspond to the region outside the range of the graph of your function.

As for why $y$ can be zero when it would make the expression undefined, this is because in order to get the expression for $x$ we had to divide by $y$, which subtly changes the function (since we can't do that if $y=0$) and we have to check separately if $y=0$ will work. It is important to watch out for this sort of slight change when manipulating functions.

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    O, i understand right now, thanks guys, sorry for my errors. :(2011-11-10