inverse of x-1/x

May 2010
4
0
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
 
Last edited:

matheagle

MHF Hall of Honor
Feb 2009
2,763
1,146
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.
 
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Apr 2009
293
94
Boston
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:

\(\displaystyle y = x - \frac{1}{x}\)

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

\(\displaystyle xy = x^2-1\)

\(\displaystyle x^2-yx - 1 = 0\)

Apply quadratic formula:

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

Good luck!
 
May 2010
4
0
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
 
Mar 2010
41
18
unfortunalty, im stuck in hicks state. IN.
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?
 
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May 2010
4
0
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.
 
Last edited:
May 2010
4
0
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?