# inverse of x-1/x

#### scottboston

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

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#### matheagle

MHF Hall of Honor
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.

• autumn

#### apcalculus

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!

#### 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

#### autumn

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?

• matheagle

#### scottboston

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.

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#### matheagle

MHF Hall of Honor
try subtracting

#### scottboston

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?