This should be a piece of cake but I really have no idea about how to do it.
I have to calculate $f^{-1}(2)$ where $f(x) = \ln(x) + 2x^5$.
I proved the function is invertible (that is: injective and surjective), but I am blocked about he request of $f^{-1}(2)$.
I cannot find a way to calculate $f^{-1}$. any help? I also have to find the derivative of the inverse function, but in this case I will be fine.
On
We don't need to solve for $f^{-1}(x)$. In fact, doing so looks like it would be yucky. So we only need to solve $f(x)=2$ which will imply $f^{-1}(2)=x$.
IN your comment you asked how to find general $f^{-1}(a)$ for a general $a$. The best way to do this is solve $f(x)=a$ from which it follows that $f^{-1}(a)=x$.
The thing not to do here is to try to find $f^{-1}(x)$ in terms of $x$ and then plug in $x=2$. Instead, note $f^{-1}(2)=b$ if and only if $f(b)=2$, and $b=1$ plainly works. Sometimes you get lucky.
In general, being able to find a closed-form expression for $f^{-1}(x)$ is rare.