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)?

1 Answer
Aug 4, 2017

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#