# Thread: inverse of x-1/x

1. ## inverse of x-1/x

Its been quite a few years since my calculus days so I'm probably missing something obvious. Somebody can probably do this in a couple seconds...

f(x)=y-(1/y)

solve for y.

many thanks

2. first of all you have a typo

you probably mean f(y)=y-1/y

in any case to solve y=x-1/x, you multiply by x giving you a quadractic

$\displaystyle xy=x^2-1$ so use the quadratic equation and solve for x.

3. Originally Posted by scottboston
Its been quite a few years since my calculus days so I'm probably missing something obvious. Somebody can do this a a couple seconds...

f(x)=y-(1/y)

solve for y.

many thanks
Not sure if you were looking for a more elegant technique here, but the quadratic formula should do the job:

$\displaystyle y = x - \frac{1}{x}$

$\displaystyle y = \frac{x^2-1}{x}$

$\displaystyle xy = x^2-1$

$\displaystyle x^2-yx - 1 = 0$

$\displaystyle x = \frac{y\pm \sqrt{y^2+4}}{2}$

Good luck!

4. Thanks for the replies

Something is up using the quadratic here because I'm getting 0=4 when I try to use the result $\displaystyle 0 = (x + \frac{y + \sqrt{y^2+4}}{2})(x + \frac{y - \sqrt{y^2+4}}{2})$

Any thoughts? thanks

5. Originally Posted by scottboston
Thanks for the replies

Something is up using the quadratic here because I'm getting 0=4 when I try to use the result $\displaystyle 0 = (x + \frac{y + \sqrt{y^2+4}}{2})(x + \frac{y - \sqrt{y^2+4}}{2})$

Any thoughts? thanks
why are you taking x plus these roots?

6. I'm just trying to solve one of them at zero

Souldn't I be trying to solve this?
$\displaystyle x = \frac{y + \sqrt{y^2+4}}{2}$

And then this?
$\displaystyle x = \frac{y - \sqrt{y^2+4}}{2}$

I know I'm missing something here...something tells me I'll be doing and infinite loop of quadratic equations.

7. try subtracting

8. ok

$\displaystyle 0 = (x - \frac{y + \sqrt{y^2+4}}{2})(x - \frac{y - \sqrt{y^2+4}}{2})$

try the first one....

$\displaystyle 0 = x - \frac{y + \sqrt{y^2+4}}{2}$

$\displaystyle -2x = -y + \sqrt{y^2+4}$

This is where I end. I don't think I want any xy multiple so I'm not sure what to do with the square root. How am I doing?