Question

Let `f(x)= -35x-x^5` and let g be the inverse function of f, how do you find a) g(0) b) g'(0) c) g(-36) d) g'(-36)?

Answer 1

Parts a, c by inspection. Parts b, d use `g'(a) = 1/(f'(g(a))`

Explanation:

For constant `a`, `g(a)` is the solution to `f(x) = a`

Parts a, b
`g(0) = 0` because `f(0) = 0`

Note that `f'(x) = -35-5x^4`, so `f'(0) = -35` and

`g'(0) = 1/(f'(g(0))) = 1/(f'(0)) = 1/(-35) = -1/35`

Parts c, d
`g(-36) = 1` because `f(1) = -36`

Note that `f'(x) = -35-5x^4`, so `f'(1) = -40` and

`g'(-36) = 1/(f'(g(-36))) = 1/(f'(1)) = 1/(-40) = -1/40`