f(x)=x^3 + 2x -1
How do I find the inverse function?
that's what I thought ... you do not need the inverse function to do this problem.
$\displaystyle f(x) = x^3 + 2x - 1$
note that $\displaystyle f(1) = 2$ , so if $\displaystyle g(x)$ is the inverse of $\displaystyle f(x)$, then $\displaystyle g(2) = 1$
further, if two functions, $\displaystyle f(x)$ and $\displaystyle g(x)$, are inverses, then $\displaystyle f[g(x)] = x$
$\displaystyle \frac{d}{dx} \left(f[g(x)] = x\right)
$
$\displaystyle f'[g(x)] \cdot g'(x) = 1$
$\displaystyle g'(x) = \frac{1}{f'[g(x)]}
$
$\displaystyle g'(2) = \frac{1}{f'[g(2)]}$
$\displaystyle g'(2) = \frac{1}{f'[g(2)]}$
$\displaystyle g'(2) = \frac{1}{f'(1)}$
find $\displaystyle f'(1)$ and finish it.
What do you mean by "any calculus like that"? If you are asked this question, surely you have worked with derivatives and the chain rule? If g is the inverse function to f, then f(g(x))= x so, by the chain rule, f'(g(x))g'(x)= 1. g(2)= 1 because $\displaystyle f(x)= x^3+ 2x- 1= 2$ has 1 as an obvious root. f'(g(2))= f'(1) which is $\displaystyle 3(1)^2+ 2= 5$. So 5g'(2)= 1, g'(2)= 1/5.