1. ## Inverse function

Hi, can someone please help me with a problem? I have to find the inverse function, but I have NO idea about how to do it. To find inverses, we always used the "make x the subject of the equation, then swap x for y" method, but this doesn't seem to work here.

"Find $f^{-1}(x)$ given that $f(x)=\frac{\arcsin{x}}{\ln{x}}$ "

Thanks.

2. Actually it does work!

For the function $f(x) = \frac{arcsin(x)}{\ln{x}}$

$y = \frac{arcsinx}{\ln{x}}$

$y\ln{x} = arcsin(x)$

$arcsin{x} =(y\ln{x})$

$y = sin (x \ln{x})$

Hope that helps you!

3. Originally Posted by mollymcf2009
$arcsin{x} =(y\ln{x})$

$y = sin (x \ln{x})$
How does this last step work. If you are just exchanging x for y then shouldn't you get

$y = sin (x \ln{y})$

?

4. I'm sorry I should not have put parentheses around that second part

The answer is y = sin(x) ln(x)

5. Originally Posted by mollymcf2009
I'm sorry I should not have put parentheses around that second part

The answer is y = sin(x) ln(x)
Sorry. I still don't see how you get that from

$\arcsin x = y \ln x$

even if you do exchange variables.

I can't see how to express an inverse function of

$y = \frac{\arcsin x}{\ln x}$

$y \ln x = \arcsin x$

$\ln x^y = \arcsin x$

$x = \sin\left(\ln x^y\right)$

$x = \frac{1}{2i}\left(x^y - x^{-y}\right)$

so basically x is the root of an equation

$x^y - 2ix - x^{-y} = 0$

which for a general $y$ has no explicit solution.

but maybe I'm missing something...

6. Originally Posted by Rincewind
Sorry. I still don't see how you get that from

$\arcsin x = y \ln x$

even if you do exchange variables.

I can't see how to express an inverse function of

$y = \frac{\arcsin x}{\ln x}$

$y \ln x = \arcsin x$

$\ln x^y = \arcsin x$

$x = \sin\left(\ln x^y\right)$

$x = \frac{1}{2i}\left(x^y - x^{-y}\right)$

so basically x is the root of an equation

$x^y - 2ix - x^{-y} = 0$

which for a general $y$ has no explicit solution.

but maybe I'm missing something...
The thing you're missing, and in fact the thing that we're all missing, is the exact question. We only have the OP's version of it .......

Here is a scenario that could lead to the posted question:

Given that $f(x) = \frac{\arcsin x}{\ln x}$ find the value of $f^{-1}\left( - \frac{\pi}{6 \ln 2}\right)$.

Or perhaps a scenario where the value of the derivative of the inverse function is asked for ....

7. I hate myself.
Since you guys said that there was something missing, I checked the question again. The exercise was written in an extremely small font size, and it turns out that it didn't ask for $f^{-1}(x)$ but for $f'(x)$, which is perfectly solvable.

Sorry (and I will bring a magnifying glass the next time I try to solve a math problem)!